Approximate Differential Equations for Renormalization Group Functions in Models Free of Vertex Divergencies
aa r X i v : . [ h e p - t h ] N ov Approximate Differential Equations forRenormalization Group Functions in Models Free ofVertex Divergencies.
Marc P. Bellon ∗ , † Abstract
I introduce an approximation scheme that allows to deduce differen-tial equations for the renormalization group β -function from a Schwinger–Dyson equation for the propagator. This approximation is proven to givethe dominant asymptotic behavior of the perturbative solution. In thesupersymmetric Wess–Zumino model and a φ scalar model which do nothave divergent vertex functions, this simple Schwinger–Dyson equation forthe propagator captures the main quantum corrections. New methods for efficient perturbative calculations in Quantum Field The-ory are badly needed, both for practical and theoretical reasons. This workexplores the possibility of approximating Schwinger–Dyson equations as differ-ential equations on the renormalization group functions in specific cases.Four situations will be studied and compared, based on two distinctions: theSchwinger–Dyson equation is either linear or quadratic and the theory is either afour dimensional supersymmetric model or a six dimensional scalar one. Thesemodels are characterized by the absence of divergences in vertex functions, at allorders for the Wess–Zumino model, only at one loop for the scalar model. Thisallows to avoid Schwinger–Dyson equations for vertex functions which wouldbe much more difficult to study.In the case of linear Schwinger–Dyson equations, Broadhurst and Kreimerobtained differential equations for the renormalization group functions in [1].A generalization to the case of non-linear Schwinger–Dyson equations is theaim of this work. In a preceding work [2], we have shown how to obtain nu-merically high orders of the perturbative solution of a nonlinear Schwinger–Dyson equation, elaborating on methods proposed in [3]. These computationsshowed the singularities of the Borel transform of the perturbative series: thesingularity on the positive axis is weaker than the one predicted by cruder ap-proximations. However the origin of the observed asymptotic behavior of the ∗ UPMC Univ Paris 06, UMR 7589, LPTHE, F-75005, Paris, France † CNRS, UMR 7589, LPTHE, F-75005, Paris, France
The present study only considers massless models where renormalization islimited to the propagator. As in [2], one of these models is the supersymmetricWess–Zumino model. The vertex functions are never divergent in this case, ascan be proven to all orders in perturbation theory by superspace techniques.Another model of interest is the six-dimensional theory of a scalar complex fieldwith the interaction: λ
3! ( φ + ¯ φ ) (1)The one-loop three-point function is zero: the vertices associated to the φ interactions and those associated to its complex conjugate must alternate andthis is not possible in a three point loop. Furthermore, the two-loop contribu-tion to the three-point function is non-planar, so that it does not contributein a large N limit. It is therefore a coherent approximation to consider theSchwinger–Dyson equation associated to its unique one-loop divergence, whichis a propagator correction.The fundamental object of study of this work is therefore the simplest non-2inear Schwinger–Dyson equation, which is graphically: (cid:16) (cid:17) − = 1 − a (2)In this equation, a denotes a suitable equivalent of the fine structure constant,which is equal to λ up to some numerical constant.The simpler linear Schwinger–Dyson equation, which has been extensivelystudied in [1], will also be considered as a test bed and for comparative purposes.It is graphically depicted as (cid:16) (cid:17) − = 1 − a (3)In both cases, these equations express a one-particle irreducible two-pointfunction in term of an integral over the propagator, which is its inverse. Thedifficulty in providing for solutions of these Schwinger–Dyson equations stemfrom the need to obtain the full propagator, when only its first derivative iseasily deduced. The gap is bridged by the use of the renormalization groupcombined with a renormalization condition taken at a fixed impulsion. The two-point functions are always considered as a ratio with respect to theirfree counterpart. Adding a renormalization condition at a fixed exterior impul-sion p = µ , the two-point functions get an expansion in power of the logarithmof the impulsion L = log( p /µ ): G ( L ) = 1 + X k g k L k k ! (4)I introduce here a factor k ! to take g k as the k th derivative of the function, aconvention which was not in either [3] or [2] but proves convenient. Now, therenormalization group relates values of the propagator at different impulsionsand allows to deduce a recursive relation between the g k . In fact, as we haveseen in [2], similar recursion relations can be found for any power, positiveor negative, of the propagator. The recursion has two parameters, the power n of the propagator we consider and a parameter b which is 2 for the linearSchwinger–Dyson equation and 3 for the non-linear one. b is the power of thepropagator appearing in the effective coupling constant: g k +1 = γ ( n + ba∂ a ) g k . (5)In this equation, γ is the anomalous dimension, the derivative of the propagatorwith respect to the variable L . I will note γ k the coefficients g k in the case ofthe propagator, with the obvious equality γ = γ .The proof, detailed in [2], is based on two elements. On the one hand,the sums of the diagrams of a given order generate a sub Hopf algebra of the3enormalization Hopf algebra [4] and this property remains true for the partialsums including only the diagrams generated by a Schwinger–Dyson equation.On the other hand, the renormalization group is a one parameter group inthe group of characters of the renormalization Hopf algebra. This fact is simplerto establish in our case than in the original works of Connes and Kreimer [5].In the minimal subtraction scheme they used, the counterterms are not algebrahomomorphism so that it is not trivial that the ratio of the counterterms at twodifferent scales gives an algebra homomorphism. Here however, renormalizationis at a fixed scale, so that the counterterm is simply the convolution inverse ofthe evaluation at the renormalization scale. The renormalization group is asimple consequence of the definition of the renormalized evaluation, if we knowthat the renormalized evaluation has a well defined limit when the regulator isremoved. Φ Rq /p = (Φ p ◦ S ) ⋆ Φ q = (Φ p ◦ S ) ⋆ Φ p ⋆ (Φ p ◦ S ) ⋆ Φ q = Φ Rp /p ⋆ Φ Rq /p In this equation, the convolution inverse is written using the antipode S of therenormalization Hopf algebra.The introduction of these Hopf algebra structures has for me two virtues.It allows to give a precise sense to the fact that the very same combinatorialidentities which make the renormalization program work are at the base ofthe renormalization group. In the case we are interested in, it shows thatthe necessary relations hold when we do not deal with the full perturbativedevelopment, but only to the part generated by a truncated Schwinger–Dysonequation. In both cases, we simply have to evaluate a single scalar loop integral. Thisis clear for the φ case, and was proven in [6, 2] for the supersymmetric one.Schwinger parameterization allows to obtain the following result: F ( a, b ) = Z dk D ( k ) a (( p − k ) ) b = π D ( p ) D − a − b Γ( D − a )Γ( a ) Γ( D − b )Γ( b ) Γ( D − c )Γ( c ) (6)with c defined as c = D − a − b . This expression is very symmetric in a , b and c , and this can be explained by the introduction of the two loop vacuumdiagram with three propagators having the exponents a , b and c . The choice of c makes this diagram scale invariant and hence divergent. This infinite resultcan however be understood as a finite quantity multiplying the infinite volumeof the dilatation group. If the invariance under the group of rotations anddilatations is fixed by choosing one of the impulsions, one obtains this finitecoefficient. The scale can be fixed in a covariant way with ( p ) D δ ( p − p ),to obtain a result independent of the impulsion which has been fixed. Thisenhanced symmetry is not very important in this simple one-loop case, where4n explicit expression for the result is at hand, but afford great simplificationsfor higher loop cases.At a given order, the propagator takes the form of the free propagatormultiplied by a polynomial in L = log( p /µ ). It can be obtained from theaction of a differential operator on ( p ) − a . More precisely, we write: P ( p ) = 1 p (cid:16) X γ k log( p /µ ) k k ! (cid:17) = (cid:16) X γ k ∂ kx k ! (cid:17) p ) − x (cid:12)(cid:12)(cid:12)(cid:12) x =0 (7)If we plug this value in the loop integral, we can exchange the derivation andthe loop integration, so that we end up multiplying the γ k by the coefficientsof the Taylor expansion around (0 ,
0) of F (1 − x, − y ).The object of interest is therefore the Taylor expansion of F (1 − x, − y ).The function F has poles whenever one of the Γ function in the numerator hasa negative integer as argument, i.e., for the parameters equal to D or greaterintegers. The residues are simple polynomials. When a or b is a great enoughinteger, it corresponds to infrared divergences. For c instead, the pole comesfrom a ultraviolet divergence, but the two are linked by conformal invariance.This link between poles of the Mellin transform and divergences of the diagramgives clues for the computation of the residues even at higher loop order, whenthe full Mellin transform has no closed form expression. This should allow tocontrol the effect of higher loop corrections to the Schwinger–Dyson equationand will be the subject of future work. In the cases with linear Schwinger–Dyson equations, differential equations forthe anomalous dimension were presented in [1]. However the process throughwhich these equations were encountered, involving explorations and computerassisted transformations of a system of partial differential equations, was notenlightening. A first explanation was presented in this same work, through thenotion of propagator–coupling duality: this gives the equations we derived fromthe renormalization group.The four dimensional case is really simple, without any possible variation,and is left as an exercise for the interested reader. In the φ case, equation (6)is taken with b = 1, a = 1 − x and D = 6. The Mellin transform becomes asimple rational function. The Schwinger–Dyson equation takes the followingform: Π( L ) = 1 − a ( X γ k ∂ kx k ! )[ e Lx −
1] 1 x (1 + x )(2 + x )(3 + x ) (8)Conformal invariance corresponds to the exchange of a and c and implies asymmetry for x transformed to − − x . The application of a suitable differentialoperator with respect to L allows to reduce the right hand side to a constant,and the different terms of the left hand side can be obtained from γ throughthe use of some version of equation (5). This derivation has been presented inYeats’ thesis [7]. 5his case will be used to test the approximation scheme I propose: let usforget the poles most removed from the origin and compensate with the firstfew terms of the Taylor expansion. One obtains a simpler differential equation.Instead of applying the full differential operator ∂ L (1 + ∂ L )(2 + ∂ L )(3 + ∂ L ),we drop the last factors and truncate the Taylor series for 1 / (2 + x )(3 + x ) toobtain a differential expression on γ of similar order. Instead of(3 + γ (2 a∂ a − γ (2 a∂ a − γ (2 a∂ a − γ = a, (9)one obtains: γ + γ (2 a∂ a − γ = a (cid:16) − γ + 1936 γ (2 a∂ a + 1) γ (cid:17) (10)Let us remark that the terms quadratic in γ on the left and right hand sides ofthis equation differ, since in one case, one is dealing with the higher derivativesof the 1PI two-point function and in the other one, with those of its inverse,the full propagator. A comparison of numerical solutions of the two equationsshow that the simplified equation (10) fairs remarkably well: the two solutionsare visually indiscernible up to a = 2, the relative error is around half of apercent at a = 1. The behaviors however diverge for larger a . When theexact equation (9) has a regular solution for large a , behaving as a / up tologarithmic corrections, the solution of the approximate equation (10) hits asingularity around a = 11 . n are multiplied by ( − n n − to get natural numbers, beginning with1, 11, 376. These first coefficients are equal, since the approximation is exactup to this order. The coefficients for the approximate solution are comparableto the one of the full solution, with a ratio decreasing slowly as n − β , with β around 0 .
22. This value stem from the computation of 200 coefficients. Onecould also study the intermediate situation where only the (3 + ∂ L ) factor isdropped, but this is of limited interest.What I wanted to show is that with the action of a suitable differential op-erator such that the Taylor coefficients of the Mellin transform become rapidlysmall, truncating the Taylor expansion to a few terms nonetheless producessensible results. Furthermore, an equation as (10) allows to compute easily theratio of two successive terms in the perturbative expansion, which is n/
3. Thiswould be less clear from the full equation (9). φ In the case of a nonlinear Schwinger–Dyson equation for the six-dimensionalmodel, the relevant Mellin transform is: e L (1+ x + y ) Γ(2 + x )Γ(1 − x ) Γ(2 + y )Γ(1 − y ) Γ( − − x − y )Γ(4 + x + y ) (11)The poles associated to the factors Γ(2 + x ) or Γ(2 + y ) are farther from theorigin and give contributions to the Taylor expansion which decrease at least6s 2 − n at order n . The dominant contributions for the high order of the Taylorexpansion come from the poles of Γ( − − x − y ). As in the preceding case, wecan differentiate with respect to L in order to cancel the poles near the origin.More precisely, we need to multiply the Mellin transform by ( − − x − y )( − x − y )(1 − x − y ) which corresponds to the action of ∂ L − ∂ L . Using the relevantrenormalization group equation for the higher derivatives of γ and a suitabletruncation of the Taylor expansion one obtains: γ − γ (3 a∂ a − γ (3 a∂ a − γ = a (cid:18) − γ + a
18 (49 γ (3 a∂ a + 1) γ + 67 γ ) (cid:19) (12)The consequences of this equation will not be detailed here, since one lackssuitable comparison points. However, it is easy to see that for large orders in a ,the dominant term in the expansion of γ comes from the cubic in γ term on theleft hand side. Associated to the lowest order of γ which is a/
6, this proves thatthe ratio of successive terms is asymptotically − n/
2. This fixes the convergenceradius of the Borel transform of the perturbative series and indicates that themain singularity is on the negative axis.
The γ function for this model was studied in [2] and a number of observationscould be made on the behavior of the resulting series. The approximations wepropose here will be checked against the detailed computations we made, andreciprocally, the approximations allow to prove the observed properties of theseries.Our starting point is the Mellin transform obtained in [2]: − ( e L ( x + y ) −
1) Γ(1 + x )Γ(1 + y )Γ( − x − y )Γ(1 − x )Γ(1 − y )Γ(2 + x + y ) (13)This case presents a new problem. This Mellin transform presents a pole for x + y = 1, but the dominant ones are the poles for x = − y = −
1. This poles cannot be cancelled by derivations with respect to L , asthe ones depending only on x + y . The residue of the pole for x = − − y or 1 + xy due to the derivation withrespect to L necessary to cancel the divergence for x = y = 0.In a first step, let us consider the contribution coming only from (1 + x ) − .The application of the differential operator P γ k ∂ kx /k ! gives the sum of the( − k γ k . Since γ k +1 is deduced from γ k by the application of the operator γ (3 a∂ a + 1), we obtain the formal series: X k ( − k γ k = X k ( − k [ γ (3 a∂ a + 1)] k γ (3 a∂ a + 1) 1 . (14)The exact definition of the inverse does not matter, since we will be multiplyby the operator to cancel it. However, since the sum of the γ k is multiplied by a , the operator must be permuted with the operation of multiplying by a . We7herefore multiply both sides of the equation by the operator 1 + γ (3 a∂ a − γ = 2 aγ − a + 2 a
11 + γ (3 a∂ a + 1) 1 (15)[1 + γ (3 a∂ a − γ + a − aγ ) = 2 a (16) γ = a − aγ − γ (3 a∂ a − γ + 2 aγ + 2 a (2 a∂ a + 1) γ (17)With this formula, it is easy to obtain the asymptotic growth of the coeffi-cients in the development of γ = P n ( − n − c n a n . The term γ (3 a∂ a − γ givesthe ratio of successive terms proportional to 3 n . The next term for this ratiois easy to compute, since the term cubic in γ and aγ do not contribute at thislevel of precision. One obtains: c n +1 ≃ (3 n + 2) c n (18)This is exactly the result obtained experimentally from the calculation in [2].The fact that this most simple approximation has this ratio asymptoticallyexact up to the constant terms is at first a surprise. It nevertheless has theinteresting consequence that the ratio of these approximate coefficients and theexact ones will reach a finite limit. Indeed, a product of terms which behavesasymptotically as 1 + O (1 /n ) is convergent. The comparison of the coefficientsobtained from the iterative solution of equation (17) with the more preciseresults obtained in [2] indeed shows that their ratio, which starts at 1 for thefirst terms, has a limit which can be estimated to be 0 . . a . The rapid growth of the coefficients of γ makes the term of order n of a product dominated by the term with one of the factors of the highestpossible order. The recursive definition of γ k than shows that the coefficientsof a n in all the γ k are of comparable size when n is larger than k . In a product γ γ k , the dominant contribution will come from the term of degree n − γ k ,which is a factor of order n smaller, and products involving γ will make stillsmaller contributions.The terms which are linear in the γ k are therefore asymptotically dominant.In the Wess–Zumino model, the pole at x = − y = 0 and therefore for this dominant terms, whereas in the φ model, twofurther poles are necessary to obtain the full contribution.We should nevertheless expect that the terms coming from xy/ (1 − x − y )and xy/ (1 + x ) contribute finite terms to equation (18), since they represent n terms of size 1 /n . However, xy/ (1 − x − y ) has all its Taylor coefficientspositive so that there are cancellations due to the alternating signs of the γ k for a given order. The cases of xy/ (1 + x ) and xy/ (1 + y ) are more subtle. Thecorresponding contributions can be written as: a γ
11 + γ (3 a∂ a + 1) γ (19)8hen they are multiplied by the operator 1 + γ (3 a∂ a − a ( γa∂ a γ ) 11 + γ (3 a∂ a + 1) γ (20)The infinite series of terms is therefore multiplied by a term of order a , so thatthe coefficients are individually proportional to c n /n , and their sum cannotcontribute a finite term. We have thus shown that the differential equation (17)allows to predict the ratio of the successive terms of the series for γ up tovanishing terms. In order to reach a higher precision on the asymptotic behavior, it is necessaryto take into account the pole for x + y = 1 and the full residue of the poles at x or y = −
1. Canceling the pole for x + y = 1 can be achieved simply by takingderivatives with respect to L , the logarithm of the impulsion, to multiply theMellin transform by ( x + y )(1 − x − y ). This however add to the complexity ofthe residue of the poles at x and y = −
1. The Mellin transform now reads:Γ(2 − x − y )Γ(1 + x )Γ(1 + y )Γ(2 + x + y )Γ(1 − x )Γ(1 − y ) =(1 − x )(2 − x )1 + y + (1 − y )(2 − y )1 + x − x + y ) (21) − ( x + y ) + 2( ζ (3) − xy ( x + y ) + · · · In our preceding work, we remarked that expressing the residues in terms ofthe product xy allowed for a simpler polynomial part, but in the present case,it is better to have the different summands of the residue give similar terms.The corresponding equation for the γ function, dropping the term proportionalto ζ (3) −
1, is: γ − γ (3 ∇ − γ = − a + 6 a γ − a γ (3 ∇ + 2) γ + a (4 − γ + 2 γ (3 ∇ + 1) γ ) 11 + γ (3 ∇ + 1) 1 (22)The operator ∇ has been introduced as a short hand for a∂ a to keep down thesize of the equation. The formal inverse can be removed by putting everythingelse on the other side: γ + 3 a − γ (3 ∇ − γ − a γ + 2 a γ (3 ∇ + 2) γ − γ + 2 γ (3 ∇ + 1) γ = 11 + γ (3 ∇ − a (23)In this form, the differential equation obtained by applying 1+ γ (3 ∇−
2) to bothsides looks rather daunting. The presence of a quotient reintroduces the neces-sity of series inversion that we avoided by a clever use of the renormalizationgroup equations. Otherwise, the expansion of the derivative of the quotient,followed by a multiplication by the square of the denominator in (23) gives apolynomial equation, but with numerous terms.9et us remark that in any case, the derivatives with respect to a get multi-plied by aγ : the total number of possible terms of a given degree in γ is thereforelimited. It is possible that the combinatorial methods of the operad of algebraswith derivation introduced by Jean-Louis Loday [8] is useful to stitch togethersimilar terms. The complexity of the obtained equation raises the questionwhether a systematic improvement of such approximations by the addition ofthe contribution of other poles of the Mellin transform is practical.The factor aγ coming with each derivatives has a double consequence. Onone side, it ensures that perturbatively, higher derivative terms are subdomi-nant, but this also makes the differential equation highly singular in the vicinityof a = 0: proving the non-perturbative existence of the solution is not straight-forward. In the present paper, I have shown how to deduce from Schwinger–Dysonequations simple differential equations for the renormalization group functions.They readily give the asymptotic behavior of the perturbative series and in par-ticular the convergence radius of the Borel transform. Through the inclusion ofthe contributions of more poles of the Mellin transform, it should be possibleto obtain systematic improvements of the solution.Differential equations for renormalization group functions had been pro-posed in recent years. In her thesis [7], Karen Yeats proposed a way to linearizethe nonlinear Schwinger–Dyson equations and obtain simple differential equa-tions for the renormalization group functions. The proposition has been appliedboth to QED [9] and to QCD [10]. However, the transformed Schwinger–Dysonequation is indeed linear, but with an infinite number of terms and an unknownfunction appears in the differential equation, with a very complex recursive def-inition. It is therefore not clear if in a perturbative solution, the contributionfrom the non-linear differential term dominates the contribution of the unknownfunction.Up to now, we only considered simple Schwinger–Dyson equations, witha one-loop correction to the propagator. However, the full Schwinger–Dysonequation includes higher order terms, and we would like to know how theseadditional contributions modify the properties of the renormalization groupfunctions. The difficulty a priori with such terms is the great number of propa-gators, each coming with its own variable, and therefore the rapid growth of thenumber of terms with a given number of derivatives of the Mellin transform.However the leading contributions in the Taylor expansion of the multivariableMellin transforms correspond to its poles, which can be related to the diver-gences of the diagram. Indeed, with every propagator coming with a variableexponent, all subdiagrams become divergent for some choice of the exponents.A rˆole should be find for the core Hopf algebra introduced in [11, 12] to organizethese divergences. The poles have a simple structure, because they only dependon the sum of the Mellin variables of a given subgraph. The highly nonlinearcharacter of such Schwinger–Dyson equations should not be a hindrance to their10uccessful use. In particular, at least in a large N limit where the number ofprimitive divergences does not grow too fast, it could be possible to show thatthese additional terms do not change the leading asymptotic behavior of theperturbative series.Another desirable extension is to deal with vertex renormalization. Howeverthe vertices depend a priori on different energy scales and the full vertex isnot entirely defined by its renormalization group dependence. There are alsooverlapping divergences, which mean that it is not possible to simply replacethe sum of a vertex and its counterterm by a renormalized vertex. We must alsochoose the renormalization point for the vertex. In QED, the Ward identitiesare simpler for the vertex with a zero impulse photon, but this is not a suitablechoice in a massless theory.Whatever the successes we encounter in these improvements, this work hasalready delivered. It has shown how to combine Schwinger–Dyson equations andrenormalization group to control the asymptotic behavior of the perturbativeseries for exactly renormalizable quantum field theories which are not so artifi-cial than the ones studied in [1, 6]. This is a remarkable result, since the simplerecursions developed here subsume huge number of individual graphs with theirhierarchy of counterterms, with important cancellations between contributionsof differing signs. Acknowledgments:
I wish to express my special thanks to Olivier Babelonwho suggested to look for global properties of the Mellin transform in order tounderstand the asymptotic properties of its local expansion which puzzled us inour preceding work. The presentation of our previous work in seminars helpedme to clarify the concepts: thanks to all who invited me and specially to theorganizers of a workshop in Carg`ese.
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