Higher Order Corrections to the Asymptotic Perturbative Solution of a Schwinger-Dyson Equation
aa r X i v : . [ h e p - t h ] M a r Higher Order Corrections to the Asymptotic PerturbativeSolution of a Schwinger–Dyson Equation.
Marc P. Bellon , , Pierre J. Clavier UPMC Univ Paris 06, UMR 7589, LPTHE, F-75005, Paris, France CNRS, UMR 7589, LPTHE, F-75005, Paris, France
Abstract
Building on our previous works on perturbative solutions to a Schwinger–Dyson for themassless Wess–Zumino model, we show how to compute 1 /n corrections to its asymptoticbehavior. The coefficients are analytically determined through a sum on all the poles of theMellin transform of the one loop diagram. We present results up to the fourth order in 1 /n aswell as a comparison with numerical results. Unexpected cancellations of zetas are observedin the solution, so that no even zetas appear and the weight of the coefficients is lower thanexpected, which suggests the existence of more structure in the theory. Keywords : Renormalization; Schwinger–Dyson equation; Borel transformation. : 81T15;81T17.
Introduction
Quantum Field Theory (QFT) is the set of mathematical tools allowing us to describe (at least)three of the four known fundamental interactions, as well as a number of phenomena in statisticalmechanics and solid state physics. Perturbative solutions are behind successes in particle physics(QED, electroweak theory and QCD, put together into the Standard Model).Nonetheless, the present state of perturbative QFT is not very satisfying since the maximumcomputed order has grown rather slowly, due to the very fast growth of the number of graphs tocompute with the order in the perturbation theory, as well as rising difficulties in their evaluations.It is therefore not clear how we may know which (nor how many) graphs of a given order will givea significant contribution. There are also physically relevant situations where the perturbationtheory breaks down, or is not efficient (e.g., low-energy QCD). Many solutions have been probedfor these issues, like lattice QCD, effective models.An other way to overcome them is to use Schwinger–Dyson equations. Although these equationscome from perturbation theory, they can also be studied non-perturbatively. The main difficultyis that one must resort to truncated versions, which may not display important properties ofthe full theory, like the Ward identities associated to gauge invariance. Nevertheless, importantqualitative properties of QCD, like spontaneous chiral symmetry breaking and color confinement,can be obtained from the numerical study of simplified Schwinger–Dyson equations (see e.g., [1]and reference therein). However, exact solutions of Schwinger–Dyson equations are only knownfor a linear one [2].Nevertheless, much can be learned about the perturbative series of quantum field theoriesthrough the solution of a particular Schwinger–Dyson equation, in the four-dimensional super-symmetric Wess–Zumino model. The breakthrough came from the recognition of a Hopf algebrastructure on the Feynman graphs of a QFT in [3], [4]. This Hopf algebra structure takes care of theusual combinatorial technicalities of QFT and set them into an algebraic framework. A key pointof this work is that the renormalisation group can be studied in this algebraic context. So one endsup with new relations on the quantities relevant for the study of the renormalisation group. Onecan plug these relations into the Schwinger–Dyson equation of the massless Wess–Zumino modelso it becomes much simpler to solve. In particular, we do not need a huge numerical study toextract results.The method that we are going to use here has been fully explained in [5] and [6]. The startingpoint of those computations was the work [7] where it was made clear how the renormalisation1roup equation can be used to deduce the full propagator from the anomalous dimension and howthe anomalous dimension of a theory can be derived from its Schwinger–Dyson equation.This paper has five parts. The first is a brief review of the renormalisation group seen in theframework of Connes–Kreimer Hopf algebra of renormalisation, the Schwinger–Dyson equation ofthe massless Wess–Zumino model and the methods and results of [5]. In the second part we definea powerful change of variables which will drastically reduce the complexity of the computations toobtain subdominant terms in the higher orders of the perturbative solution. The results of [5] aregiven with the new set of variables. In the third part we write down the Schwinger–Dyson equationwith the first contributions from every poles of the Mellin transform (which we will define later)and solve them. The fourth part is devoted to higher order computations, up to the fourth order.This fourth order is found to have an unexpected behaviour, as explained at the very end of thisfourth part. Finally, we compare our result to the numerical results of [8]. An excellent agreementis found and the two last unconstrained parameters of our method are numerically determined.
In a given massless QFT, one can expand the two point point function in power of the logarithmof the impulsion L = ln( p /µ ). G ( L ) = 1 + + ∞ X k =1 γ k L k k ! . (1)The γ ’s are themselves functions of the fine structure constant of the theory, which we will denoteby a . γ k = + ∞ X n =0 γ k ; n a n . (2)As proven in the thesis [9] and in the article [8] the renormalisation group yields a recursion relationon the γ k ’s: γ k +1 = ( γ + βa∂ a ) γ k . (3)Here β is the β -function of the theory and ∂ a the derivative with respect to a . This result stemsfrom the work of Connes and Kreimer [4], where the renormalisation group was shown to be a oneparameter subgroup of the group of characters of the Hopf algebra of the Feynman graphs, as wellas the existence of a sub-Hopf algebra generated by the coefficients of the Green functions [10, 8].Then the equation (3) comes from the fact that the β -function is a derivative.Now, in the massless Wess–Zumino model, the vertices are not divergent and therefore do notneed to be renormalized. Only the propagator is affected by the renormalisation group and, thanksto the supersymmetry, all components of the supermultiplet get the same renormalisation factor.The Callan–Symanzik equation then leads to β = 3 γ . The proof of this result is detailed in [11]. Hence the recursion on the γ ’s is very simple. γ k +1 = γ (1 + 3 a∂ a ) γ k . (4)In this model, all the coefficients of the expansion of G ( L ) are therefore simple functions of thefirst coefficient. In the massless Wess–Zumino model, the only important Schwinger–Dyson equation for the com-putation of renormalization group functions is therefore the one for the propagator and we takeits first approximation, which is the simplest non-linear one and can be graphically described by: (cid:16) (cid:17) − = 1 − a . (5)2his simplest equation is non trivial to solve since the loop integral depends on the unknownpropagator. However, at a given loop order, the full propagator is the free propagator times thetwo-point function, which has a finite expansion in the logarithm of the momentum. P ( p /µ ) = 1 p /µ + ∞ X k =1 γ k L k k ! ! (6)The next move is to take the Mellin transform of this loop integral. This is nothing but noticing: (cid:18) ln p µ (cid:19) k = (cid:18) ddx (cid:19) k (cid:18) p µ (cid:19) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 . (7)Then, after an exchange between sum, derivatives and integral, one has only one integral (a Mellintransform) to perform.I( q /µ , x, y ) = g π Z d p p /µ ) − x [( q − p ) /µ ] − y = a (cid:18) q µ (cid:19) x + y Γ( − x − y )Γ(1 + x )Γ(1 + y )Γ(2 + x + y )Γ(1 − x )Γ(1 − y ) (8)with a = g / π and Γ being Euler’s gamma function. We will work with the derivative of thisintegral, which will give γ , H ( x, y ) = − ∂ I( q /µ , x, y ) ∂L (cid:12)(cid:12)(cid:12)(cid:12) L =0 (9)Writing (cid:16) q /µ (cid:17) x + y = exp(( x + y ) L ) one finds: H ( x, y ) = a Γ(1 − x − y )Γ(1 + x )Γ(1 + y )Γ(2 + x + y )Γ(1 − x )Γ(1 − y ) , (10)which is nicely symmetric in x and y and finite around the origin. To write the Schwinger–Dysonequation in a compact way, let us define the transform I : C [[ x ; y ]] → C [ a ] by: I ( f ( x, y )) = + ∞ X n =1 γ n n ! d n dx n ! + ∞ X m =1 γ m m ! d m dy m ! f ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) x = y =0 . (11)The non-linear Schwinger–Dyson equation (5) then gives: γ = I ( H ( x, y )) (12)In the following, we will call this γ the anomalous dimension of the theory and denote it simplyby γ . In equation (12), the number of different terms contributing at a given order grows quadraticallywith this order, making an asymptotic analysis unpractical. The solution proposed in [5] is to firstapproximate the complicated function H ( x, y ) by its pole parts with a suitable extension of theirresidues. The contributions of every pole to the anomalous dimension can then be computed.The function H ( x, y ) has poles at x ; y = − k , k ∈ N ∗ (we call these poles the simple ones)and at the lines x + y = + k , k ∈ N (the general poles). The simple poles are linked with the IRdivergences of the loop integral while the general poles come from its UV divergences. Both kindsof pole arise when, in the Mellin transform, a subgraph becomes scale invariant [6].Now, the simple poles can be expanded as:1 k + x = 1 k + ∞ X n =0 (cid:16) − xk (cid:17) n , (13)so that by (12), the contribution of such a pole to the Mellin transform is: F k = 1 k + ∞ X n =0 (cid:18) − k (cid:19) n γ n (14)3ith the convention γ = 1. Using the recurrence relation (3) between γ n and γ n +1 , one obtains:( γ + βa∂ a ) F k = − kF k + 1 . (15)For the other poles, the situation is slightly more subtle. The full proof is given in [5] butbasically the numerators, i.e., the residues of H ( x, y ) at those poles, must be taken in accountfrom the start. We will show later that the residues of H ( x, y ) are polynomials. Let us call Q k ( x, y ) the residue of H ( x, y ) at x + y = k . Then the numerator at this pole is: N k ( ∂ L , ∂ L ) = Q k ( ∂ L ∂ L ) . (16)The contribution L k of this pole to the Mellin transform cannot be expressed simply in terms ofthe γ n as in (14), but it can be computed from the following equation that it obeys:( k − γ − βa∂ a ) L k = N k ( ∂ L , ∂ L ) G ( L ) G ( L ) | L = L =0 . (17)In [5], the function H ( x, y ) was approximated by its first poles at x = − y = − x + y = +1, giving the following approximating function in (12):1 a h ( x, y ) = (1 + xy ) (cid:18)
11 + x + 11 + y − (cid:19) + 12 xy − x − y + 12 xy. (18)This means that we only use the contributions F ≡ F of the poles 1 / (1 + x ) and 1 / (1 + y ) and L ≡ L of the pole xy/ (1 − x − y ) to compute γ . Then the renormalisation group equations(15) and (17) for F and L and the Schwinger–Dyson equation (12) with the approximate function h ( x, y ) defined in (18), gives the three coupled non-linear differential equations: F = 1 − γ (3 a∂ a + 1) F,L = γ + γ (3 a∂ a + 2) L,γ = 2 aF − a − aγ ( F −
1) + a ( L − γ ) . (19)We look for a perturbative solution of these equations, and expand F , L and γ in powers of a : F = P f n a n , L = P l n a n and γ = P c n a n . With the assumption that the { f n } , the { l n } and the { c n } have a fast growth and keeping only the dominant contributions, one obtains three coupledrecursions formulas, which can be solved to obtain the two dominant terms in each series: f n +1 ≃ − (3 n + 5) f n ,l n +1 ≃ nl n ,c n +1 ≃ − (3 n + 2) c n . (20)We will see later that this results are the base for an ansatz for a systematic improvement of theseasymptotic results while taking into account every pole of H ( x, y ).For now, let us just say that the results (20) nicely fit with the numerical study of [8]. Now, one would like compute the 1 /n corrections to the asymptotic solution (20). However thiskind of computation turns out to be quickly tedious. To simplify our calculations, we separate thealternating contributions to ( c n ) from the ones with a constant sign. We will define two symbols forthis, one to encode the asymptotic behavior coming from l n and the other one for the asymptoticbehavior coming from f n . They will be defined through two series: ( A n +1 = − (3 n + 5) A n B n +1 = 3 nB n (21)but we will only use the following two symbols corresponding to the formal series: ( A = P A n a n B = P B n a n (22)4he relations (21) can be expressed as differential equations on the symbols A and B : ( a ∂ a A = − A − aA a ∂ a B = B (23)In fact, (21) do not entirely determine A and B . We must have an initial condition at an order n .Then (23) is true up to a term of degree n . Therefore, n shall be chosen higher than the degreeto which we will compute the gamma function.Although these relations are not totally exact they will drastically reduce the complexity ofthe computation of the corrections to the asymptotic behavior, allowing us to go up to the fourthorder, with computations of the same degree of complexity that those of the second order with theprevious method. To do that, we will define nine unknown functions, which are the coefficients of A and B for the F , L and γ functions. F = f + Ag + BhL = l + Am + Bnγ = a ( c + Ad + Be ) (24)The a in the ansatz for γ comes from the a in the function H ( x, y ). In what follows, c = c ( a ) willbe called the low order part of γ .Now, one can rewrite the Schwinger–Dyson equations (19) in the language of this new set offunctions, with the derivatives of the A and B symbols being removed, thanks to the relations (23).The new equations are found by saying that the coefficient of the symbol A and the one of thesymbol B shall independently vanish, since the two divergences of the Mellin transform are ofdifferent nature and thereof do not talk to each other. Similarly, the term without any symbolshould also independently vanish. Hence, this change of variables allows to efficiently separate thealternating part and the part of constant sign of the Mellin transform. It will drastically simplifythe equations to solve since we will have three equations for each of the previous ones, with theoverall system being of the same complexity than the one with the old formalism.One then ends up with the nine equations: f + c (3 a ∂ a + a ) f = 1 (25a) g + d (3 a ∂ a + a ) f + c ( − − a + 3 a ∂ a ) g = 0 (25b) h + e (3 a ∂ a + a ) f + c (1 + a + 3 a ∂ a ) h = 0 (25c) l = a c + ac (3 a∂ a + 2) l (25d) m = 2 a dc + c (3 a ∂ a − − a ) m + ad (3 a∂ a + 2) l (25e) n = 2 a ec + c (3 a ∂ a + 1 + 2 a ) n + ae (3 a∂ a + 2) l (25f) c = 2 f − − ac ( f − −
12 ( l + a c ) (25g) d = 2 g + 12 am + 2 ad (1 − f ) − ac (2 g + ad ) (25h) e = 2 h + 12 an + 2 ae (1 − f ) − ae (2 f + ac ) (25i)There are obviously more terms into the expansion of the equations (20), proportional to A , AB , B , but they will not be considered: if A and B begin by a large number of vanishingcoefficients, they correspond to corrections of very high order.The perturbative solution of these equations goes as follows. We write: c ( a ) = + ∞ X n =0 c n a n , (26)and similarly for all the other functions. At each order, the equations should be solved in the rightorder: one shall first solve the equations for f and l , then for c , then for g , m and h , n , and finally5or d and e . Following this procedure, one ends up with the solution up to the order a . f ( a ) = 1 − ag ( a ) = g + g ah ( a ) = − an l ( a ) = 0 m ( a ) = 0 n ( a ) = n + n ac ( a ) = 1 − ad ( a ) = 2 g + ( − g + 2 g ) ae ( a ) = 12 n + 12 ( n − n ) a Here the assumption of fast growth of the series which was done in [5] is not necessary since thesymbols A and B take care of the necessary properties.The coefficients g and n are not specified at this stage. It is a general feature of thisparametrization that one needs to go at the order a p +1 to fix the parameters g p and n p . Indeed,since c = 1, the a n order in the equation (25b) is: g n + ( ... ) − c g n + ( ... ) = 0 . therefore does not depend on g n , with a similar phenomenon appearing in (25f). However, the nextorder of equations (25b) and (25f) is not hard and does not involve higher coefficients of d or e , sothat we obtain the solution (up to the order a ) to the equations (19) with only two unconstrainedparameters. We however need the values of the next order for c . f ( a ) = 1 − a (27a) g ( a ) = g (cid:18) a (cid:19) (27b) h ( a ) = − an (27c) l ( a ) = 0 (27d) m ( a ) = 0 (27e) n ( a ) = n (cid:18) − a (cid:19) (27f) c ( a ) = 1 − a (27g) d ( a ) = 2 g (cid:18) a (cid:19) (27h) e ( a ) = 12 n (cid:18) − a (cid:19) (27i)The fact that there remain two unconstrained parameters, g and n , is not really surprisingsince they were already present in the former formalism, where the asymptotic behavior was inferredfrom the ratio of successive coefficients of the Taylor series. Since only ratios could be computed,the overall factors in the asymptotic behavior of the series for F and L are unconstrained. In thisnew formalism, equations stemming from the part linear in A are linear in the coefficients d , g and m of A in the unknown functions: if there is any non trivial solution, all its multiples are alsosolutions. If we had not used the analysis of the previous work [5], the precise recurrence for A could have be obtained from the requirement of the existence of a non trivial solution. Analogousstatements hold for the terms proportional to B .Up to now, we have only obtained the first two coefficients of every functions, since the otherpoles of the H ( x, y ) function will contribute to the next terms. This is the subject of the nextparts. 6 Higher poles of the Mellin transform
To go further we must include the contributions of all the poles of the Mellin transform H ( x, y ) tothe γ function. The equations for F and L do not change. Then we have to compute the residuesof H ( x, y ) at its various poles. They are given for k ≥ H, x = − k ) = − yk − k Y i =1 (cid:16) − yi (cid:17) k − Y i =1 (cid:16) − yi (cid:17) , Res(
H, x = k − y ) = ( k − y ) yk ( k + 1) k − Y i =1 (cid:18) − ( k − y ) yi ( k − i ) (cid:19) , with the convention Q k − i =1 = 1 for k = 2. In order to simplify the computations, we use the factthat the first polynomial is defined at x = − k and the second at x = k − y to make the numeratorssymmetric in x and y . One gets:Res( H, x = − k ) = xyk ( k − k Y i =1 (cid:16) xyki (cid:17) k − Y i =1 (cid:16) xyki (cid:17) = P k ( xy ) (28)Res( H, x = k − y ) = xyk ( k + 1) k − Y i =1 (cid:18) − xyi ( k − i ) (cid:19) = Q k ( xy ) . (29)The coefficients of those polynomials will be of interest. Hence we define them in the followingway: P k ( X ) = k − X n =1 p k,n X n Q k ( X ) = k X n =1 q k,n X n . Notice that the residues at the poles in y = − k are exactly the same since H ( x, y ) is symmetricunder the exchange of x and y . We therefore write:1 a H ( x, y ) = (1 + xy ) (cid:18)
11 + x + 11 + y − (cid:19) + 12 xy − x − y + + ∞ X k =2 (cid:18) k + x + 1 k + y − k (cid:19) P k ( xy ) + + ∞ X k =2 Q k ( xy ) k − x − y + ˜ H ( x, y ) . (30)The − /k term coming with the poles at x = − k and y = − k does not contribute to the singularitiesbut appears necessary to obtain the exact Taylor expansion of H ( x, y ) around the origin. Moreover,˜ H ( x, y ) shall be a holomorphic function and is the difference between H ( x, y ) such as written in(10) and the above expension of sums over the poles. We have checked that ˜ H ( x, y ) shall be ofdegree at least 10. We have also verified that some infinite families of derivatives of ˜ H ( x, y ) vanishat the origin. Hence we will make the conjecture ˜ H ( x, y ) = 0 in the following, which is the raisond’ˆetre of the − /k . We are fairly confident that this conjecture is true, but it would be pleasantto have a proof of it, which would give a stronger ground to our computations.To obtain the anomalous dimension of the theory at a given order p , one must include additionalterms of the Schwinger–Dyson equations to deal with all the contributions at this order. Indeed,we have seen in section 1.3 that the equations for L k and γ depend on the residues of H ( x, y ).Since those residues are polynomial, and because of the definition of the transformation I , we cantruncate those equations to take care only of the terms which will contribute to a given order.Let us write P k,p ( X ) for the polynomial P k ( X ) truncated to a degree less or equal to p , and Q k,p ( X ) for the polynomial Q k ( X ) similarly truncated. Then the equation (17) for L k becomes( k − γ − γa∂ a ) L k = Q k,p − ( ∂ L ∂ L ) G ( L ) G ( L ) | L = L =0 , (31)7nd the right hand side of the equation for γ in (19) gets an additional part in its right hand side, a + ∞ X k =2 L k + a + ∞ X k =2 I (cid:18) P k,p − ( xy ) (cid:20) k + x − k (cid:21)(cid:19) (32)The 2 in the equation for γ is there because for each k , H ( x, y ) has a pole x = − k and at y = − k which give equal contributions. Moreover P k,p ( X ) is just the polynomial P k ( X ) truncated to adegree less or equal to p , and similarly for Q k,p ( X ). I is the linear transform defined in section 1,which reduced in this pole expansion to: ( I (cid:16) ( xy ) n k + x (cid:17) = ( − k ) n γ n h F k − k P n − i =0 (cid:0) − k (cid:1) i γ i i , I (( xy ) n ) = γ n . (33)We have used: ( xy ) n k + x = ( − k ) n y n " x + k − k n − X i =0 (cid:16) − xk (cid:17) i . With this definition of I we see that only the term ( xy ) p − is needed for the solution at the order a p +1 of γ since the leading term of a I (( xy ) p ) which is aγ n is of order 2 p + 1. However, whenlooking at the coefficients of the symbols A and B in γ n , they still are proportional to a . The term a ( xy ) p will therefore contribute terms of order a p +2 .Last, but not least, the equation for F k is similar to the one for F , and is given in (15). Thisequation will never change, whatever the order one needs, simply because any new term whichmight affect F k will come through changes in γ . The modifications to the L k functions insteadcome also from changes to the equation of L k . To start our study of the effect of the infinitude of poles, we will take only the ( xy ) contributionto each pole. Hence the Schwinger–Dyson equation comes from the following approximate Mellintransform:1 a h ( x, y ) = (1 + xy ) (cid:18)
11 + x + 11 + y − (cid:19) + 12 xy − x − y + + ∞ X k =2 (cid:18) k + x + 1 k + y − k (cid:19) xyk ( k −
1) + + ∞ X k =2 k − x − y xyk ( k + 1) . (34)In the equation for L k we use only the linear term for the numerator in (17):[ k − γ (2 + 3 a∂ a )] L k = 1 k ( k + 1) ∂ L ∂ L G ( L ) G ( L ) | L = L =0 = 1 k ( k + 1) γ For the Schwinger–Dyson equation, one has simply to apply Eq. (12). Some series arise, whichare easily computable. So we end up with five coupled non-linear partial differential equations tosolve, F = 1 − γ (3 a∂ a + 1) F (35a) L = γ + γ (3 a∂ a + 2) L (35b) kF k = 1 − γ (1 + 3 a∂ a ) F k (35c) kL k = 1 k ( k + 1) γ + γ (2 + 3 a∂ a ) L k (35d) γ = 2 aF − a − aγ ( F −
1) + 12 aL + 2 aγ − aγ [3 − ζ (2)] + a + ∞ X k =2 (cid:18) L k − γ F k k − (cid:19) (35e)with ζ being Riemann’s zeta function. One may be worried by the ζ (2) in the last equation.Indeed, using the link between the logarithm of the Euler’s gamma function, the Riemann’s zeta8unction and the Euler–Mascheroni’s constant γ :ln Γ( z + 1) = − γz + + ∞ X k =2 ( − k k ζ ( k ) z k (for example in [12]), it is easy to see that one can write the exact Mellin transform as an exponentialof a sum over the odd values of the Riemann’s zeta function: H ( x, y ) = a x + y exp (cid:16) + ∞ X k =1 ζ (2 k + 1)2 k + 1 (cid:0) ( x + y ) k +1 − x k +1 − y k +1 (cid:1)(cid:17) . (36)The Mellin transform was already written in this form in [8], so one expects only odd zeta valuesin the result. However, the sum will give compensating terms and this ζ (2) will not appear anymore in the result. This provides a check that the calculations are correct.Now, as in Section 2, we can define the functions f k , g k and h k , and l k , m k and n k forthe functions F k and L k . Then the system of equations (35a)-(35e) shall be rewritten for thosefunctions. One ends up with fifteen coupled partial non-linear differential equations for fifteenfunctions, that we will not write down explicitly.Solving those equations should be done in the same order than in the section 1, with theequations for f k and l k solved with the equations for f and l , and similarly those for g k , m k , h k and n k with h and m . As in the previous case, the order two terms of n ( a ) and g ( a ) are not fixedby the a equations. However we only need the order three terms of the equations for g and n to fixthis, while the most tedious equations to solve at a given order are those for d and e which involvesums over k . Fixing those two last coefficients thus does not add much complexity. Moreover, wedo not need to add more terms in the γ equation, since we are looking for the equation on thecoefficient c of γ , and the higher order (such as the ( xy ) term) will act on the d and e terms only,thanks to the relations (23). The already computed orders a and a are unchanged by the additionof the new terms as expected and, all computations being done, we end up with the solution tothe equations (35a)-(35e) up to the order a . f ( a ) = 1 − a + 6 a (37a) g ( a ) = g (cid:18) a + 29 [ −
65 + 12 ζ (3)] a (cid:19) (37b) h ( a ) = n (cid:18) − a + 2912 a (cid:19) (37c) l ( a ) = a (37d) m ( a ) = 2 a g (37e) n ( a ) = n (cid:18) − a + 89 [28 − ζ (3)] a (cid:19) (37f) f k ( a ) = 1 k − k a + 2(2 + k ) k a (37g) g k ( a ) = − g k ( k − (cid:18) a + 12 − k + 13 k k ( k − a (cid:19) (37h) h k ( a ) = n k ( k + 1) (cid:18) − a k + 7 k k ( k + 1) a (cid:19) (37i) l k ( a ) = a k ( k + 1) (37j) m k ( a ) = 4 g k ( k + 1) a (37k) n k ( a ) = n ( k − k ( k + 1) a (37l) c ( a ) = 1 − a + 14 a (37m) d ( a ) = 2 g (cid:18) a + 29 [ −
71 + 12 ζ (3)] a (cid:19) (37n) e ( a ) = n (cid:18) − a + 16 (cid:20) − ζ (3) (cid:21) a (cid:19) (37o)9ith the n and g being fixed by a computation at the a order. g = 29 [ −
65 + 12 ζ (3)] g n = 89 [28 − ζ (3)] n So this order is a nice check of our procedure since the two first order are unchanged and ζ (2)disappears everywhere as expected. Now, we feel confident about the method and we will reachthe fourth order. a order We first have to determine the coefficients of P k, ( X ), which appears in the equation for γ . It issimply: P k, ( X ) = Xk ( k −
1) + X k ( k −
1) ( H k + H k − ) . (38)This is true for all values of k with the convention that H k , the k th harmonic number, is definedby H = 0, H k = H k − + 1 /k . Then the equation for γ becomes: γ = 2 aF − a − aγ ( F −
1) + 12 aL + a + ∞ X k =2 (cid:18) L k − γ F k k − γ F k H k + H k − k − (cid:19) + 2 aγ x − aγ [3 − ζ (2)] − aγ + 2 aγ γ (cid:2) − ζ (2) − ζ (3) (cid:3) − aγ (cid:2) − ζ (2) − ζ (4) − ζ (3) (cid:3) . (39)The only other equation to be changed is the one for L k which gets a new term Q k, ( X ) = 1 k ( k + 1) X − H k − k ( k + 1) X and so the equation for L k is now: kL k = γ (3 a∂ a + 2) L k + 1 k ( k + 1) γ − H k − k ( k + 1) γ . (40)These equations (together with the three equations for the other functions) can now be solved atthe third order. For the sake of readability, we will not write the fifteen functions at this order,but only the functions which are a part of γ . One ends up with a solution without any ζ (2 n ), c ( a ) = 1 − a + 14 a + 16 [ ζ (3) − a (41a) d ( a ) = g (cid:18) a + (cid:20) − ζ (3) (cid:21) a + 49 (cid:20) − ζ (3) (cid:21) a (cid:19) (41b) e ( a ) = n (cid:18) − a + 16 (cid:20) − ζ (3) (cid:21) a + 19 (cid:20) − ζ (3) (cid:21) a (cid:19) (41c)We have already included the coefficients g and n . g = g (cid:20) − ζ (3) (cid:21) n = n
81 [ − ζ (3)]Again, the disappearance of every even zeta values from the final result is a very useful fact. Itprovides us a check of our computations. We use the same notations to denote the functions g k ( a ), appearing as factors of A in F k and the coefficients g i of the function g ( a ), in order to keep a strong parallelism between the expansion of F and F k , and similarly for n k ( a ) and n i . We hope that the context make the two different usages clear. .2 The a order For the a order, we need to compute the coefficient of degree two in a product of linear terms.We use: n Y i =1 (1 + α i X ) = 1 + X n X i =1 α i + X h n X i =1 α i i − n X i =1 α i ! + O ( X ) . One finds easily the coefficient of the cubic term of the P k ( X ) polynomials (28) p k, = 1 k ( k − (cid:18) H k H k − + X ≤ i 01 and B 01, which correspond to fits on three values on acombination of A , B and aA , without imposing the relation we deduced between the two termsproportional to A . One clearly see the convergence improvement when using more terms of theasymptotic series. This can be seen as a check of our computations by numerical experiment.We numerically get 2 g ≃ − . n ≃ . g than on n , which was expected since the A n sequence growths faster than the B n one. Toimprove the precision on g and n , one can either go to higher order in a or compute additionalterms in the asymptotic expansion. Had we not have the numerical results of [8], we probablycould obtain the same precision on g and n with fewer low order terms of γ and some additionalterms of the asymptotic behavior, for a smaller total computational cost. This is not so important13igure 1: 2 g from different fits.here where computations remain manageable, but could be of serious interest when adding higherloop corrections to the Schwinger–Dyson equation. Conclusion In this work we have gone further on the path defined in [5]. The contributions of all the polesof the Mellin transform could be worked out, allowing us first to recover the first orders of theperturbative solution, but also to reach some non-perturbative information about a QFT withouttoo many heavy computations.At the level of the nature of the coefficients obtained, we observe that the sum over the polesdoes not produce zeta values with weights growing as twice the order, as could be expected. Thefinal result seems compatible with a higher weight of the coefficients of d n or e n of n in a systemwhere ζ (2 p + 1) has weight 2 p . Uncovering the precise mechanisms through which the highestweight terms cancel each others, how all the multiple zeta values stemming from the different sumshopefully conspire to give only products of ζ -values for odd integers looks like a combinatorialnightmare in search of a conceptual solution.We have been able to compute the a order of the γ function around 0. The dominant asymp-totic behaviors of the perturbative series encoded by A and B give rise to the first two singularitiesof the Borel transform of γ . These singularities are (cid:0) ξ + (cid:1) − / from A and ln (cid:0) ξ − (cid:1) from B .Multiplying a function by a corresponds to taking a primitive of its Borel transform, so that theterms of d and e we computed give the first derivatives of an analytic function multiplying thesebasic singularities to give the full singularity of the Borel transform at the point − / / β through a Borel sum. There may be howeveradditional singularities of the Borel transform. In fact, similar computations should allow thedetermination of the other possible singularities of the Borel transform, but the overall factorsshould be much more difficult to obtain, since this would entail a determination of the analyticextension of the Borel transform beyond its convergence disk.14igure 2: n from different fits.An essential ingredient of our computations has been the possibility of expressing the Mellintransform of the diagram as a sum over poles, which allowed us to sum up infinite numbers of termsin its Taylor expansion which have contributions of the same order to the asymptotic behavior.However, the way in which the residues are extended to all values of the variables is not fullyjustified: the sum over the poles becomes convergent and defines a meromorphic function. Thusthe true Mellin transform can differs from this meromorphic function only by an entire functionwhich vanishes at the origin together with an infinite number of its derivatives. We have no proofthat it is exactly zero. Such a result would be interesting on many accounts: it would give a firmerground to our computations, it would imply an infinite number of relations between multizetavalues through the comparison of the expansion in poles and the one in terms of ζ values at oddintegers (36). Finally, should it generalize to higher loop graphs, it would be a very interesting wayof evaluating Mellin transforms of multi loop graphs, since residues are also in these cases explicitpolynomials.In this work, we have only been up to the order a in γ . There is no reason not to go beyond,but the increasing technicalities of the computation. Moreover, there is a lack of a full table ofrelations among multi zeta values which makes the calculations quite lengthy. Such a difficultymight be overturned by a careful combinatorial analysis of the quantities arising in our problems,and is left for further investigation.An other interesting trail to follow would be the study of higher loop corrections of theSchwinger–Dyson equation, furthering the study of [6]. Indeed the next term is a three loopterm, which therefore modifies the coefficient c , and through it, the coefficients d and e . Evena large N limit would involve a four loop primitive divergence, so that this analysis must be com-plete to take into account those higher loop terms. The fact that our method has a fairly lowcomputational cost will be very interesting in this next step. Indeed with the three (or more) loopsterms in the Schwinger–Dyson equation, the number of propagators will increase rapidly, but thenumber of poles with differing contributions in our method does not grow so fast.Moreover, we have worked in a model where the only relevant Schwinger–Dyson equation forthe renormalization group is only for the propagator. In the more physically relevant case ofYang–Mills theories, one also has to deal with Schwinger–Dyson equation for vertices, and massiveparticles. But this involves many new difficulties that we hope to address one by one.15 eferences [1] Adnan Bashir, Lei Chang, Ian C. Cloet, Bruno El-Bennich, Yu-Xin Liu, Craig D. Roberts,and Peter C. Tandy. Collective Perspective on Advances in Dyson–Schwinger Equation QCD. Communications in Theoretical Physics , 58:79–134, July 2012.[2] David J. Broadhurst and Dirk Kreimer. Exact solutions of Dyson–Schwinger equations foriterated one-loop integrals and propagator-coupling duality. Nucl. 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