An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams
aa r X i v : . [ m a t h . C O ] O c t An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams
Ali Assem Mahmoud
Abstract.
We derive a functional relation between the generating functions of connected chord dia-grams and 2-connected chord diagrams. This relation enables us to calculate an asymptotic expansionfor the number of 2-connected chord diagrams on n chords. The asymptotic information obtained fromthis expansion refines the last established results and provides a simple alternative for calculating theasymptotic behaviour of certain Green functions in Quenched QED and Yukawa theory in the contextof quantum field theory.
1. Introduction k -Connected chord diagrams (also known in the literature as k -irreducible linked diagrams) havebeen studied in many research areas including combinatorics, quantum field theory and bioinformatics[
16, 33, 20 ], and, in particular, this research is motivated by the related applications in quenchedquantum electrodynamics (QQED) in [ ] and which the author develops here and in [ ]. Namely, theinteger sequence 1 , , , , , , . . . appeared in [ ] and [ ] as the coefficients of some renormal-ization counterterms in QQED. Unlike [ ], the asymptotic analysis presented here is used to calculatethe asymptotic behaviour of these counterterms without the need of singularity analysis, and dependingonly on the combinatorial interpretation.In this paper we shall study the asymptotic behaviour of the number of 2-connected chord diagrams.Informally speaking, these are chord diagrams which require the removal of at least two chords to getthem disconnected. Here we obtain an asymptotic expansion for ( C > ) n , the number of 2-connectedchord diagrams on n chords. As we have mentioned earlier, in [ ], it is shown that the proportionof connected chord diagrams approaches e − as the number of chords goes to infinity. The work ofStein and Everett in [ ] addresses a special case of a more general result by Kleitman in [ ], wherethe argument was less detailed. Kleitman argues that the proportion of k -connected chord diagramsgoes to e − k . In [ ], M. Borinsky showed that the asymptotic behaviour of connected chord diagrams isapproximated by a series expansion, in which the first term corresponds to the e − obtained by Steinand Everett, and earlier by Kleitman; whereas the infinitely many extra terms provide higher precisionas needed. Our result here will extend Kleitman’s result in very much the same way, this time for thecase of 2-connected chord diagrams. Namely, we obtain an asymptotic expansion for 2-connected chorddiagrams, in which the first term corresponds to the e − in Kleitman’s argument. However, to be ableto extract such information about this class of chord diagrams, we will need to work on producing arecursion that relates 2-connected chord diagrams with connected chord diagrams. k -Connected Chord Diagrams Definition . A chord diagram on n chords (i.e. of size n ) is geometricallyperceived as a circle with 2 n nodes that are matched into disjoint pairs, with each pair correspondingto a chord . Definition . A rooted chord diagram is a chord diagram with aselected node. The selected node is called the root vertex , and the chord with the root vertex is calledthe root chord . In other words, a rooted chord diagram of size n is a matching of the set { , . . . , n } . For an algebraic definition, this is the same as a fixed-point free involution in S n . Then the generatingseries for rooted chord diagrams is(2.1) D ( x ) := X n =0 (2 n − x n All chord diagrams considered here are going to be rooted and so, when we say a chord diagram wetacitly mean a rooted one.Now, a rooted chord diagram can be represented in a linear order, by numbering the nodes incounterclockwise order, starting from the root which receives the label ‘1’. A chord in the diagram maybe referred to as c = { a < b } , where a and b are the nodes in the linear order. Definition . In the linear representation of a rooted chord diagram, an interval isthe space to the right of one of the nodes in the linear representation. Thus, a rooted diagram on n chords has 2 n intervals.For example, this includes the space to the right of the last node (in the linear order). intervalsthe root the root Figure 1.
A rooted chord diagram and its linear representationAs may be expected by now, the crossings in a chord diagram encode much of the structure and sowe ought to give proper notation for them. Namely, in the linear order, two chords c = { v < v } and c = { w < w } are said to cross if v < w < v < w or w < v < w < v . Tracing all the crossingsin the diagram leads to the following definition: Definition . Given a (rooted) chord diagram D on n chords, considerthe following graph G D : the chords of the diagram will serve as vertices for the new graph, and thereis an edge between the two vertices c = { v < v } and c = { w < w } if v < w < v < w or w < v < w < v , i.e. if the chords cross each other. The graph so constructed is called the intersection graph of the given chord diagram. Remark . A labelling for the intersection graph can be obtained as follows: give the label 1 tothe root chord; order the components obtained if the root is removed according to the order of the firstvertex of each of them in the linear representation, say the components are C , . . . , C n ; and then recur-sively label each of the components. It is easily verified that a rooted chord diagram can be uniquelyrecovered from its labelled intersection graph. Definition . A (rooted) chord diagram is said to be connected ifits intersection graph is connected (in the graph-theoretic sense). A connected component of a diagramis a subset of chords which itself forms a connected chord diagram. The term root component will referto the connected component containing the root chord.
Example . The diagram D below is a connected chord diagram in linear representation, wherethe root node is drawn in black. N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 3 D The generating function for connected chord diagrams (in the number of chords) is denoted by C ( x ).Thus C ( x ) = P n =0 C n x n , where C n is the number of connected chord diagrams on n chords. The firstterms of C ( x ) are found to be C ( x ) = x + x + 4 x + 27 x + 248 x + · · · ;the reader may refer to OEIS sequence A000699 for more coefficients. The next lemma lists some classicdecompositions for chord diagrams (see [ ] for example). Lemma . If D ( x ) , C ( x ) are the generating series for chord diagrams and connected chord dia-grams respectively, then (i) D ( x ) = 1 + C ( xD ( x ) ) , (ii) D ( x ) = 1 + xD ( x ) + 2 x D ′ ( x ) , and (iii) 2 xC ( x ) C ′ ( x ) = C ( x )(1 + C ( x )) − x . Proof.
We sketch the underlying decompositions as follows:(i) The ‘one’ term is for the empty chord diagram. Now, given a nonempty chord diagram, wesee that for every chord in the root component there live two chord diagrams to the right ofits two ends. This gives the desired decomposition.(ii) There are three situations for a root chord: it is either non-existent (empty diagram); or it isconcatenated with a following diagram; or the root chord has its right end landing in one ofthe intervals of a diagram. These situations correspond respectively with the terms in (ii).(iii) Can be derived from (i) and (ii). Nevertheless, it can be also shown as follows: if we remove theroot chord what is left is a sequence of connected components, with each component having aspecial interval (through wich the root used to pass) which cannot be the last interval (see thefigure below). Thus each of these components is counted according to the generating function2 xC ′ ( x ) − C ( x ).This decomposition gives that C ( x ) = x − (2 xC ′ ( x ) − C ( x )) , and the result follows. (cid:3) The main object we use throughout is chord diagrams with certain degrees (strengths) of connec-tivity.
Definition k -Connected Chord Diagrams) . A chord diagram on n chords is said to be k - connected if there is no set S of consecutive endpoints, with | S | < n − k , S is paired with lessthan k endpoints not in S (here we assume the endpoints are consecutive in the sense of the linearrepresentation). In other words, the diagram requires the deletion of at least k chords to becomedisconnected. A k - connected diagram which is not k + 1- connected will be said to have connectivity k . Example . The diagram in Figure 2 is 3-connected since it can not be disconnected with theremoval of fewer than 3 chords, but it is not 4-connected.
ALI ASSEM MAHMOUD
Figure 2.
A diagram that is 3-connected but not 4-connected
Definition k ) . Given a connectivity- k diagram, a set ofsize k of chords is called a cut if its removal disconnects the diagram. Equivalently, a set T of k chordsin a connectivity- k diagram is a cut if there exists a sequence S of consecutive end points such that | S | < n − k and all the end points in S are paired together except for k endpoints from the k chordsin T . Such a sequence S will be called a reason for connectivity- k . See Figure 3 below for illustration. Figure 3.Notation . For the generating functions we shall use the following notation: C > k ( x ) (or C > k )will denote k -connected diagrams whereas C k ( x ) (or C k ) denotes diagrams with connectivity k . So forexample C ( x ) = C ( x ) + C > ( x ).A computation of the first coefficients gives C ( x ) = x + x + 4 x + 27 x + 248 x + 2830 x + · · · C ( x ) = x + 3 x + 20 x + 185 x + 2101 x + · · · C > ( x ) = x + x + 7 x + 63 x + 729 x + · · · (2.2)
3. Functional Recurrence for -Connected Diagrams In [ ], the (classic) functional relation D ( x ) = 1 + C ( xD ) provided the suitable grounds forderiving information about the asymptotic behaviour of C n , the number of connected chord diagramson n chords. The composition of maps in the second term in this relation transforms nicely into aproduct when taking the alien derivative A / (see Appendix A for definitions). In the aftermath of ourmeeting in the Canadian Mathematical Society session about chord diagrams (Dec. 2019), M. Borinskysuggested to the author that it may be possible to obtain similar functional relations for the higherconnectivity diagrams. This was motivated by the asymptotic pattern shown in Kleitman’s results [ ].In this section we derive such a functional relation for 2-connected chord diagrams, and will use it laterto study the asymptotic behaviour of the number of 2-connected chord diagrams. However, just as thecase for general graphs, it is not clear whether 3-connected diagrams and k - connected diagrams ingeneral do follow similar relations. Proposition . The following functional relation between connected and -connected diagramsholds: (3.1) C = C x − C > (cid:18) C x (cid:19) . N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 5
Proof.
Assume that a connected chord diagram C is given. We can determine the maximalsequences of consecutive end points that are reasons for connectivity-1. A sequence S = s s . . . s m ofconsecutive end points is of this type if and only if(1) S is a reason of connectivity-1 corresponding to a cut chord c that has exactly one end pointinside S , say this is s i where 1 < i < m .(2) S is not contained in any other reason for connectivity-1.However, these reasons for connectivity-1 may overlap (see Figure 4), and so we will need to devisea canonical way for partitioning our diagram in terms of these maximal sequences. Figure 4.
Case 1: C is the single chord diagram . In this case we do nothing, and the contribution to thegenerating function is just x .In the next cases we generally assume C is not the single chord diagram. Case 2:
The root endpoint r (left endpoint of the root chord of C ) is not contained in any reasonfor connectivity-1. In this case we determine the maximal reasons for connectivity-1 that are obtainedthrough the next procedure by moving from left to right. Such a diagram generally looks like theexample in Figure 5 below. Figure 5.
An illustration for Case 2Consider the diagram C × obtained from C as follows:(1) Starting from the left, determine the first endpoint that is included in some reason for connectivity-1. Let’s denote it temporarily by s . Move to step 5 if the diagram is 2-connected and no suchendpoint exists.(2) Determine the maximal reason S = s s . . . s m for connectivity-1 that contains s by consecu-tively trying to include the next endpoints to the right. Assume S corresponds to a cut chord c that has the end point s i , say.(3) Let C × be the diagram obtained by removing the sub-diagram induced by S − s i , i.e. weremove S without removing c .(4) Update by setting C = C × , and go back to step 1.(5) Output C × . ALI ASSEM MAHMOUD
Observation 1:
Notice that in the process of extracting C × the diagram remains connected, thisis because any of the removed sub-diagrams has been connected to the rest of C through a single cutchord which is not removed.Clearly, C × will not preserve any original reason for connectivity-1 in C . Moreover, notice thatagain since each sub-diagram removed has only been connected to the rest of C through a single cutchord (which is kept), the process should not affect the connectivity of the rest of C neither will createnew cuts. Observation 2:
Also, step 1 is exclusive throughout the procedure. Indeed, if there is no suchendpoint in a connected diagram (Observation 1) then the diagram is either 2-connected or is the singlechord diagram (it can’t be empty). The latter however never occurs: Initially the diagram is not thesingle chord diagram by our assumption. Further, C is not reduced to a single chord diagram at anyiteration since this should imply that the root endpoint r is contained in a reason for connectivity-1.Therefore the procedure eventually halts and the output is 2-connected. Observation 3:
It is important to note that also the last endpoint in C is not included in anyreason for connectivity-1, for this will imply the same for r .To summarize the procedure above, we are removing maximal reasons of connectivity-1 that appearin a certain order when moving from left to right, without removing their corresponding cuts. This isillustrated in Figures 6 and 7.This gives a reversible decomposition into a 2-connected where each endpoint, except the first andlast endpoints, is assigned to a connected chord diagram counted by one less chord. In other words,we will count each middle chord (i.e. whose endpoints are not the root nor the last endpoint) in C × when counting the connected diagram for its right endpoint by keeping it as a root for this diagram,while on the other hand, the diagram for the left endpoint will be counted by one less chord to avoidovercounting.This can also be viewed as follows:Given a connected chord diagram C (which is not the single chord) we undergo the describedprocedure to get(1) a 2-connected chord diagram C × ,(2) the root chord c r corresponds to a connected chord diagram that consists of c r and the diagramattached to the right endpoint of the root, in which we will keep the root.(3) the chord c l carrying the last endpoint of C corresponds to a connected chord diagram thatconsists of c l kept as a root for whatever the diagram attached to the left endpoint.(4) Every middle chord c can be replaced with a pair of diagrams corresponding to right and leftendpoints. The diagram for the left endpoint has its root a copy of c that is not going to becounted and is connected; while the diagram for the right endpoint keeps c and is connectedas well.In terms of generating functions the contribution of Case 2 is seen now to be:(3.2) C ( x ) " C > ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) t = C ( x ) /x , where we divide by t to account for the fact that two of the chords are treated differently (namely c r and c l ). Each of these two chords contributes with C ( x ) as shown above. Case 3:
The root endpoint r (left endpoint of the root chord of C ) is contained in a reason forconnectivity-1. In this case we determine the maximal reason for connectivity-1 containing r , dontedby S , by consecutively checking every endpoint to the right of r . Let c ∗ be the corresponding cutfor S . Now, by the maximality of S it must be that none of the reasons for connectivity-1 that comelater could be extended to contain S . This means that the diagram obtained by removing S (withoutremoving c ∗ ) is of the type considered in Case 2 above. The diagram will generally be structured as inFigure 8. Then the contribution to the generating function is N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 7
Figure 6.
A diagram C with cuts highlighted (red). C × Figure 7.
Maximal reasons obtained through the procedure (underlined), C × (bold).(3.3) C ( x ) − xx . C ( x ) " C > ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) t = C ( x ) /x , where the factor of C ( x ) − xx corresponds to the sub-diagram induced by S together with c r : the( C − x ) since S is always nonempty in this case, and we divide by x since c r is counted with the rest ofthe diagram. Figure 8. S is the maximal reason containing r , and so the rest of the diagram shouldbe covered with a 2-connected sub-diagram (blue). ALI ASSEM MAHMOUD
Thus, by combining the findings of the three cases we have C ( x ) = x + C ( x ) " C > ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) t = C ( x ) /x + C ( x ) − xx . C ( x ) " C > ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) t = C ( x ) /x = x + C ( x ) x . " C > ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) t = C ( x ) /x = x + xC ( x ) . (cid:18) C ( x ) x (cid:19) . " C > ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) t = C ( x ) /x = x + xC ( x ) . C > (cid:18) C ( x ) x (cid:19) , and the result now follows. (cid:3) For future reference, we include the first terms of the expressions involved in the previous decompo-sition. The reader can check that the sum of x plus lines 3 and 4 in the next table gives the first termsof C ( x ). x x x x x x x C /x C > ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) t = C ( x ) /x C ( x ) (cid:2) C > ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) t = C ( x ) /x (cid:3) C ( x ) − xx . C ( x ) (cid:2) C > ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) t = C ( x ) /x (cid:3) Table 1.
The first coefficients of the series involved in the terms of the decomposition of C > .
4. Asymptotics of the number of 2-connected chord diagrams
In this section we will see how to successfully estimate the number ( C > ) n of 2-connected diagramswhen n is large. The asymptotic behaviour obtained here will extend Kleitman’s result [ ] and willshed light on an unexplained pattern for the images of the alien derivative. It turns out that A C > takes the form of a rational function in C > times the exponential of a quadratic expression in thereciprocal of that rational function. This was exactly the same case for A C (as well as monolithicdiagrams and simple permutations). We will proceed now by applying a suitable alien derivative as wasdone before for connected chord diagrams.In the previous section we have seen that C = C x − C > (cid:18) C x (cid:19) . We will start by applying the alien derivative A , which is allowed since C ( x ) ∈ R [[ x ]] ⊂ R [[ x ]] by Corollary A.4. N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 9 (cid:0) A C (cid:1) ( x ) = A (cid:18) C ( x ) x (cid:19) − A (cid:18) C > (cid:18) C ( x ) x (cid:19)(cid:19) ( x )= A (cid:0) C ( x ) (cid:1) − A (cid:18) C > (cid:18) C ( x ) x (cid:19)(cid:19) ( x )= 2 C (cid:0) A C (cid:1) ( x ) − A (cid:18) C > (cid:18) C ( x ) x (cid:19)(cid:19) ( x ) , by the linearity of A and by Proposition A.5. Rearrange and use Proposition A.3 to get(2 C − x ) (cid:0) A C (cid:1) ( x ) = A (cid:18) C > (cid:18) C ( x ) x (cid:19)(cid:19) ( x ) . Now to get rid of the decomposition on the right we appeal to Theorem A.7:(2 C − x ) (cid:0) A C (cid:1) ( x ) = C ′ > (cid:18) C x (cid:19) A (cid:18) C x (cid:19) + (cid:18) x C (cid:19) e C /x − x xC /x (cid:16) A C > (cid:17) (cid:18) C x (cid:19) = 2 C (cid:0) A C (cid:1) ( x ) C ′ > (cid:18) C x (cid:19) + x C e C /x − x xC /x (cid:16) A C > (cid:17) (cid:18) C x (cid:19) . Equation (3.1) and Lemma 2.1 give that C ′ = 2 xCC ′ − C x (cid:20) − C ′ > (cid:18) C x (cid:19)(cid:21) = C + C − x − C x (cid:20) − C ′ > (cid:18) C x (cid:19)(cid:21) = C − xx (cid:20) − C ′ > (cid:18) C x (cid:19)(cid:21) . Substituting into our equation we get (cid:18) x CC ′ C − x − x (cid:19) (cid:0) A C (cid:1) ( x ) = x C e C /x − x xC /x (cid:16) A C > (cid:17) (cid:18) C x (cid:19) . Now, by [ ], (cid:0) A C (cid:1) ( x ) = 1 √ π xC ( x ) e − x ( C +2 C ) , and hence (cid:16) A C > (cid:17) (cid:18) C x (cid:19) = C x (cid:18) x CC ′ C − x − x (cid:19) (cid:0) A C (cid:1) ( x ) e x − C /x xC /x = C x · x ( C + C − x ) − xC + x C − x · (cid:0) A C (cid:1) ( x ) · e x − C /x xC /x = C x · xC C − x · √ π xC ( x ) e − x ( C +2 C ) · e x − C /x xC /x . Since the
LHS is a function in C x , applying Proposition A.3 gives that (cid:16) A C > (cid:17) (cid:18) C x (cid:19) = C x · (cid:16) A C > (cid:17) (cid:18) C x (cid:19) . Back to our equation, we thus have(4.1) (cid:16) A C > (cid:17) (cid:18) C ( x ) x (cid:19) = 1 √ π · C C − x · e − x ( C +2 C ) · e x − C /x xC /x = 1 √ π · C C − x · e − x [ C +2 C +1 − x C ] . Since the power series C ( x ) x is invertible, we let y ( x ) be such that C ( y ) y = x . In that case equation(3.1) gives C ( y ) = x − C > ( x ) , and hence y ( x ) = ( x − C > ( x )) x . Substituting y ( x ) for x in equation (4.1) we get (cid:16) A C > (cid:17) ( x ) = 1 √ π · ( x − C > ) (cid:18) ( x − C > ) − ( x − C > ) x (cid:19) · e − (cid:20) x + x ( x − C > + x ( x − C > − x (cid:21) = 1 √ π · x ( x − C > )( x − x + C > ) · e − x (cid:20) x + x (1 − C > /x ) + − C > /x )2 − (cid:21) = 1 √ π · x (cid:18) C > (1 − C > /x ) (cid:19) · e − x (cid:20)(cid:18) − C > /x ) + x (cid:19) − (cid:21) . In other words,(4.2) (cid:16) A C > (cid:17) ( x ) = 1 √ π · x C > S · e − x [ ( S + x ) − ] , where S ( x ) = 1 (cid:16) − C > ( x ) x (cid:17) is the generating series for sequences of 2-connected chord diagramscounted by one less chord.Finally it is noteworthy to see that the image A C > of C > ( x ) under the alien derivative is ofthe form of a rational function of C > times the exponential of a quadratic expression in the rationalfunction. The same pattern has been observed in the case of connected chord diagrams. From anotherpoint of view, one can see that xS ( x ) also counts connectivity-1 diagrams in which only the root chordis a cut.The evaluation of A C > will enable us to derive information about the asymptotic behaviourwhich strongly extend the result by Kleitman in [ ]. First let us list the first few terms of the functionsinvolved.Note that we are willing to display the factor of e − that comes from exp (cid:8) − x (cid:2) ( S + x ) − (cid:3) (cid:9) andthat is why the last row in Table 2 is multiplied by e . N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 11 x x x x x x x x S ( x ) = 1 / (cid:16) − C > ( x ) x (cid:17) S + x ) x (cid:2) ( S + x ) − (cid:3) C > · S x C > · S e · exp (cid:8) − x (cid:2) ( S + x ) − (cid:3) (cid:9) − − − Table 2.
The first terms of the series involved in calculating A C > ( x ).The computation then gives (cid:16) A C > (cid:17) ( x ) = 1 √ π · x C > S · e − x [ ( S + x ) − ]= e − √ π (cid:2) − x − x − x − x − x − · · · (cid:3) (4.3)Now, by Definition A.1 of factorially divergent power series and Definition A.2 of the alien derivative A , and since Γ ( n ) = √ π (2 n − R ∈ N , the number ( C > ) n of 2-connecteddiagrams on n chords satisfies( C > ) n = R − X k =0 [ x k ] (cid:16) A C > (cid:17) ( x ) · Γ ( n − k ) + O (Γ ( n − R ))= √ π R − X k =0 [ x k ] (cid:16) A C > (cid:17) ( x ) · (2( n − k ) − O ((2( n − R ) − , and hence the first few terms in this asymptotic expansion are given by( C > ) n = e − (cid:18) (2 n − − n − − n − − n − −− n − − n − − · · · (cid:19) = e − (2 n − (cid:18) − n − − n − n − − n − n − n − −− n − n − n − n − − n − · · · (2 n − − · · · (cid:19) . (4.4)The result by Kleitman [ ] corresponds to the first term in this expansion. By the above approach,any precision can be achieved and an arbitrary number of terms can be produced.Equation 4.4 also shows that a randomly chosen chord diagram on n chords is 2-connected with aprobability of 1 e (cid:16) − n (cid:17) + O (1 /n ) . In the next section we will see that this expansion also corresponds to the asymptotics of thenumber of skeleton quenched QED vertex diagrams [ ]. In that context the first five terms of the above expansion were conjectured by D. J. Broadhurst on a numerical evidence (see page 38 in [ ]) in studyingzero-dimensional field theory.
5. Connection with Zero-Dimensional QFT
In the next part of the paper we will see that some of the integer sequences produced in studying 2-connected chord diagrams appear in the context of zero-dimensional quantum field theory. Note that inthis situation the partition function transforms into a series of graphs since no actual Feynman integralshall remain. On another level, the Feynman rules will be represented as a character from H to thealgebra R [[ ~ ]] [ ]. We managed to establish the relation between 2-connected chord diagrams and someof the observables in quenched QED . In [ ] the asymptotics for these sequences are obtained througha singularity analysis approach. We will be able to get the same asymptotics through an enumerativeapproach. Factorially divergent power series are, as expected, used in both approaches, and hence wewill regularly appeal to theorems from Section A. First we will briefly set-up the context in perturbationtheory. In most parts we follow the notation in [ ].Recall the basic path integral formulation of QFT and notice that for zero-dimensional QFT thepath integral for the partition function becomes an ordinary integral given for example by Z ( ~ ) := Z R √ π ~ e ~ (cid:16) − x a + V ( x ) (cid:17) dx, where V ( x ) ∈ x R [[ x ]] is the potential and the exponent − x a + V ( x ) is the action and denoted by S .As known, this integral generally has singularities, and even as a series expansion it generally have asingularity at zero. In [ ], the expansion is treated as a formal power series and the focus is on studyingthe asymptotics of the coefficients.Recall that Gaussian integrals satisfy Z R √ π ~ e − x a ~ x n dx = √ a ( a ~ ) n (2 n − , were only the even powers are considered since the integral vanishes for odd powers. This enables us towork with a well-defined power series instead of the path integral (actually this is the path integral indimension 0): Definition ]) . For a general formal action S ( x ) = − x a + V ( x ) ∈ x R [[ x ]] we define thecorresponding perturbative partition function to be the power series in ~ given by F [ S ( x )]( ~ ) = √ a ∞ X n =0 ( a ~ ) n (2 n − x n ] e ~ V ( x ) . This is a well-defined power series in ~ since the coefficient [ x n ] e ~ V ( x ) is a polynomial in ~ − ofdegree less than n because V ∈ x R [[ x ]]. Just as the path integral, this map also has a diagrammaticmeaning in terms of Feynman diagrams [ ]: Proposition . If S ( x ) = − x a + P ∞ k =3 λ k k ! x k with a > , then F [ S ( x )]( ~ ) = √ a X Γ ~ | E (Γ) |−| V (Γ) | a | E (Γ) | Π v ∈ V (Γ) λ n v | Aut Γ | , where the sum runs over all multigraphs Γ in which the valency n v of every vertex v is at least ,and where | E (Γ) | , | V (Γ) | , and | Aut Γ | are the sizes of the edge set, the vertex set, and the automorphismgroup of Γ respectively. So, in terms of Feynman diagrams, to compute the n th coefficient of F [ S ( x )] we do the following(1) Draw all multigraphs Γ with | E (Γ) | − | V (Γ) | = n . Note that this is one less than the loopnumber, it is the number of independent cycles in the graph (remember that independentcycles can be obtained by starting with a spanning tree and adding one edge at a time). Theloop number is also known as the Betti number of the graph. The number | E (Γ) | − | V (Γ) | willbe referred to as the excess of Γ. N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 13 (2) Each vertex contributes with a factor that corresponds to its valency, that’s how we getΠ v ∈ V (Γ) λ n v . Then we multiply with the factor a | E (Γ) | . This process simply corresponds to theFeynman rules. The map that applies the Feynman rules will be denoted by φ S .(3) Divide by the size of the automorphism group of the graph.(4) Finally sum up all the contributions and multiply by √ a . Example ]) . As an example, the action for ϕ -theory takes the form S ( x ) = − x + x . Inthat case Z ϕ ( ~ ) = ∞ X n =0 ~ n (2 n − x n ] e x ~ = ∞ X n =0 (6 n − n (2 n )! ~ n . In terms of Feynman diagrams only 3-regular graphs will show up in ϕ -theory, hence we have Z ϕ ( ~ ) = φ S (cid:16) · · · (cid:17) . Then applying the Feynman rules φ S does not change the coefficients in the sum since the contri-bution of any vertex is 1 according to the described potential. Adding up the terms with the same loopnumber then gives Z ϕ ( ~ ) = 1 + 524 ~ + 3851152 ~ + · · · , which agrees with the first algebraic calculation. We will study such expansionsthat arise in QED theories, namely we shall consider quenched QED and Yukawa theory. These areexamples of theories with interaction. It is impossible to completely cover the underlying physics,nevertheless we should be able to understand as much as needed for our purposes by anticipating theinterrelations between the different entities defined.In the presence of interaction in the theory, the partition function takes the form Z ( ~ , j ) := Z R √ π ~ e ~ (cid:16) − x + V ( x )+ xj (cid:17) dx, where an additional term is added to the potential, namely xj , j is called the source .With this extra term we can not directly expand the integral as we did before, but we can stillachieve the same essence after a change of variables. Shift x to x + x o where x o ( j ) is the unique powerseries solution to x o ( j ) = V ′ ( x o ( j )) + j . Then we get Z ( ~ , j ) = Z R √ π ~ e ~ (cid:18) − ( x + xo )22 + V ( x + x o )+( x + x o ) j (cid:19) dx = e ~ (cid:18) − x o + V ( x o )+ x o j (cid:19) Z R √ π ~ e ~ (cid:16) − x + V ( x + x o ) − V ( x o ) − xV ′ ( x o ) (cid:17) dx = e ~ (cid:18) − x o + V ( x o )+ x o j (cid:19) F (cid:20) − x V ( x + x o ) − V ( x o ) − xV ′ ( x o ) (cid:21) ( ~ ) . The exponential factor enumerates forests (collections of trees) with the corresponding conditionson vertices, these diagrams are referred to as the tree-level diagrams. Tree-level diagrams contributewith negative powers of ~ , and therefore we are going to isolate them so that the treatment for the mainexpansion remains clear. Remember that Feynman diagrams are labeled, and so in order to restrictourselves to connected diagrams we have to take the logarithm of the partition function: W ( ~ , j ) : = ~ log Z ( ~ , j )= − x o V ( x o ) + x o j + ~ log F (cid:20) − x V ( x + x o ) − V ( x o ) − xV ′ ( x o ) (cid:21) ( ~ )generates all connected diagrams and is called the free energy . Note that the extra ~ factor causesthe powers to express the number of loops instead of the excess. Again we are using the notation in [ ]since we are eventually going to compare to parts of the work.As customary in QFT, to move to the quantum effective action G , which generates diagrams,one takes the Legendre transform of W : G ( ~ , ϕ c ) := W − jϕ c , (5.1)where ϕ c := ∂ j W . The coefficients [ ϕ nc ] G are called the (proper) Green functions of the theory. Recallthat from a graph theoretic point-of-view, being (1-particle irreducible) is merely another wayof saying 2-connected. Thus, combinatorially, the Legendre transform, as in [ ], is seen to be thetransportation from connected diagrams to 2-connected or diagrams. In that sense, the order ofthe derivative ∂ nϕ c G | ϕ c =0 = [ ϕ nc ] G determines the number of external legs.In the next part of the discussion we shall need the following physical jargon and terminology:(1) The Green function [ ϕ c ] G = ∂ ϕ c G | ϕ c =0 generates all diagrams with exactly one externalleg, which are called the tadpoles of the theory (Figure 9). Figure 9.
A tadpole diagram in QED(2) The Green function [ ϕ c ] G = ∂ ϕ c G | ϕ c =0 generates all diagrams with two external legs.Such a diagram is called a (can replace an edge in the theory). Figure 10.
A propagator diagram(3) For n > ∂ nϕ c G | ϕ c =0 = [ ϕ nc ] G is called the n -point function .In quenched QED, some of the quantities that we are going to compare their expansions with thegenerating series of 2-connected chord diagrams are the renormalized Green functions with respect toa chosen residue . We shall therefore recall from Section B.4 the basics of the Hopf-algebraic treatmentof renormalization in the next section before proceeding into the real calculations. For more about thistopic the reader can consult [ ], or the original paper by D. Kreimer and A. Connes [ ]. N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 15
By the work in [
26, 15, 19, 21, 22, 9 ] we know that,for a given QFT theory, the superficially divergent 1PI Feynman graphs form a Hopf algebra H . Theproduct is defined to be the disjoint union, and the coproduct of a connected Feynman graph Γ wasdefined according to Definition B.19:∆(Γ) = X γ ⊆ Γ γ product of divergent1 P I subgraphs γ ⊗ Γ /γ, and extended as an algebra morphism. The unit, counit, and antipode were denoted by I , ˆ I , and S .In [ ] (and Section 3.1 in [ ]) we can see how the Dyson-Schwinger equations can be written interms of the elements X r , where X r was defined as X r = 1 ± X r , where the negative sign is assumed only when r is edge-type. Also from [ ] it is shown that if weuse insertions in case of a theory with a single vertex type we get the equation: X r = 1 ± X k B γ r,k + ( X r Q k ) , where the sum is over all primitive 1PI diagrams with loop number k and residue r , and where Q is the invariant charge as defined in Section B.7.The identity(5.2) ∆ X r = ∞ X L =0 X r Q L ⊗ X r (cid:12)(cid:12) L , is of most importance in the context of renormalization [ ]. The | L , as used in [ ], is the restrictionof the sum to graphs with loop number L .We have seen in Section B.4 that the Feynman rules are simply characters from H to a commutativealgebra A . For zero-dimensional field theories the Feynman rules will be φ : H −→ R [[ ~ ]]:(5.3) φ { Γ } ( ~ ) = ~ ℓ (Γ) , where we follow the notation in [ ] for putting the arguments from H in curly brackets.In that case, the Green functions, or the generating function of 1 P I
Feynman graphs with residue r are defined as(5.4) g r ( ~ ) := φ { X r } ( ~ ) = 1 ± X r ~ ℓ (Γ) Sym Γ , where ℓ (Γ) is the loop number as before. If residue r is the k external legs residue, then g r = ∂ kϕ c G | ϕ c =0 , the k th derivative of the quantum effective action.In our case of zero-dimensional QFT, the fact that the target algebra for the Feynman rules is R [[ ~ ]]limits the choice for a Rota-Baxter operator R : R [[ ~ ]] −→ R [[ ~ ]] that respects the grading of H . Theonly choice for a meaningful renormalization scheme in this case is R = id (see [ ]).Thus, by our definitions in Section B.4 (equation B.1), the counterterm map for the renormalizationscheme R = id is given by S φ = R ◦ φ ◦ S = φ ◦ S. Then the renormalized Feynman rules is Remember that a Rota-Baxter operator R is used in the renormalization scheme to extract (in terms of an inducedBirkhoff decomposition) the divergent part of the integral. See Section B.4. φ ren := S φR ∗ φ = S φ ∗ φ = ( φ ◦ S ) ∗ φ, (5.5)where ∗ is the convolution product (Definition B.7). However, the action of the last expression on anarbitrary element of H is:(( φ ◦ S ) ∗ φ )(Γ) = m ◦ (( φ ◦ S ) ⊗ φ ) ◦ ∆(Γ)= X γ ⊆ Γ γ product of divergent1 P I subgraphs φ ( S ( γ )) φ (Γ /γ )= φ (cid:16) X S ( γ )(Γ /γ ) (cid:17) (since φ is a character)= φ ( I (ˆ I (Γ)) (by definition of the antipode S ),which is zero for all nonempty elements in H since I ◦ ˆ I maps all elements in H to zero except for theempty graph, which is mapped to itself.Thus, equation (5.5) becomes(5.6) φ ren = I ◦ ˆ I , and whence the renormalized Green functions are(5.7) g r ren = φ ren { X r } ( ~ ) = 1 ± . Note that in [ ] the signs are different since, as mentioned earlier, X r in their convention is − z r := S φ { X r } . Notice that since H is commutative, the definition of the convolution product together with equation(5.2) now yield 1 = φ ren { X r } ( ~ )(5.9) = ( m ◦ ( S φ ⊗ φ ) ◦ ∆ X r )( ~ )= ( m ◦ ( φ ⊗ S φ ) ◦ ∆ X r )( ~ )= ( m ◦ ( φ ⊗ S φ ) ◦ ( ∞ X L =0 X r Q L ⊗ X r (cid:12)(cid:12) L ))( ~ )= ∞ X L =0 φ { X r } ( ~ ) ( φ { Q } ( ~ )) L [ ~ L ] S φ { X r } ( ~ )= φ { X r } ( ~ ) S φ { X r } ( ~ φ { Q } ( ~ ))= g r ( ~ ) z r ( ~ α ( ~ )) . Let ~ ( y ) be the unique power series solution of y = ~ ( y ) α ( ~ ( y )). In [ ], y is called the renormalizedexpansion parameter and is denoted by ~ ren . Then, substituting in equation we get(5.10) z r ( ~ ren ) = 1 g r ( ~ ( ~ ren )) . The following result was proven by M. Borinsky in [ ], and we shall depend on it in the combinatorialtreatment in the next section. Theorem ]) . In a theory with a cubic vertex-type, the numeric coefficients in z r ( ~ ren ) countthe number of primitive diagrams if r is vertex-type. N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 17
The two theories that we areconcerned with here are quenched QED and Yukawa theory, which are examples of QED-type the-ories. In these theories we have two particles: fermion and boson (wiggly and dashed edges) parti-cles, and we have only three-valent vertices of the type fermion-fermion-boson. We will compute theasymptotics of z φ c | ψ c | ( ~ ren ) in quenched QED, as well as the asymptotics of the green functions ∂ iφ c ( ∂ ψ c ∂ ¯ ψ c ) j G Yuk (cid:12)(cid:12)(cid:12) φ c = ψ c =0 . Our approach is completely combinatorial and depends on establishing bi-jections between the diagrams in the combinatorial interpretation of the considered series and differentclasses of chord diagrams. Unlike the approach applied in [ ], we do not need to refer to singularityanalysis nor the representation of S ( x ) by affine hyperelliptic curves. The partition function takes the form(5.11) Z ( ~ , j, η ) = Z R √ π ~ e ~ (cid:18) − x + jx + | η | − x + ~ log − x (cid:19) dx. We are not going to discuss the physical reasoning behind the above expression, the reader may referto QFT books or surveys for more details, e.g. see [ ]. We only hint that, combinatorially, ~ log − x generates fermion loops, while | η | − x generates a fermion propagator. The special examples of Yukawatheory and quenched QED will be as follows:(1) Quenched QED is an approximation of QED where fermion loops are not present. So thatthe term ~ log − x does not appear in the partition function. Thus, the partition function forquenched QED is given by Z QQED ( ~ , j, η ) = Z R √ π ~ e ~ (cid:18) − x + jx + | η | − x (cid:19) dx. (2) For zero-dimensional Yukawa theory the partition function is just the integral in equation(5.11). That is, the partition function for zero-dimensional Yukawa theory is given by Z Y uk ( ~ , j, η ) = Z R √ π ~ e ~ (cid:18) − x + jx + | η | − x + ~ log − x (cid:19) dx.
6. Quenched QED
For this theory we are interested in the asymptotics of the counterterm z φ c | ψ c | ( ~ ren ) obtainedin [ ] (page 38). By Theorem 5.2, since QQED has only one type of vertices which is three-valent,this series enumerates the number of primitive quenched QED diagrams with vertex-type residue (seesequence A049464 of the OEIS for the first entries).Thus, this is the same as counting the number of all diagrams γ with the following specifications:(1) two types of edges, fermion and boson (photon) edges, represented as and ,respectively;(2) only three-valent vertices with the structure , with one fermion in, one fermionout, and one photon;(3) no fermion loops;(4) the residue res( γ ) is vertex-type; and(5) γ is 1 P I primitive, in other words it is edge-connected and contains no subdivergences (Defi-nition B.20 and Definition B.18).We let Q be the class of all such diagrams. Example . The following diagram in Figure 11 is in Q , whereas the diagrams in Figure 12 arenot in Q . Figure 11.
An example of a quenched QED diagram in Q Figure 12.
Diagrams not in Q .In Figure 12, the second diagram is not 1PI, the rest of the first four diagrams are all 1PI, butthey are not primitive. The last two diagrams are not in Q and are not even quenched as they containfermion loops. Theorem . The generating series z φ c | ψ c | ( ~ ren ) and z | ψ c | ( ~ ren ) count -connected chorddiagrams. More precisely, [ ~ n − ren ] z φ c | ψ c | ( ~ ren ) = [ ~ n ren ] z | ψ c | ( ~ ren ) = [ x n ] C > ( x ) . Proof.
As we mentioned above, the class of diagrams counted by z φ c | ψ c | ( ~ ren ) is to be denotedby Q . Thus, Q consists of 1PI primitive quenched QED diagrams with two external fermion legs andone photon leg. By definition, every graph in Q has a (wiggly) photon external leg r , and two directedfermion external legs f and f . We now start by proving the following claim: Claim 1:
If Γ is a graph in Q , then there exists a unique fermion-only path P from f to f .Moreover, P passes through the vertex at r and every vertex in the graph is on P . In addition, the loopnumber in Γ is equal to the number of internal photon edges, and so either is counted by the power of ~ ren in the series. Proof:
Generally, if we remove all photon edges from a graph with the 3-valent vertex residue weshould get a single directed path of fermion edges (because otherwise we will have more than 2 external
N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 19 fermion legs if the photon edges are restored) and a set fermion loops. Now, in our case, we can onlyget the path, which we denote by P and which should then carry all the vertices in the original graph.In particular, the number of vertices in a graph Γ ∈ Q will be 1+the number of internal fermion edges.To see that the rest of the claim is indeed true first recall Euler’s formula | V (Γ) | − | E (Γ) | + ℓ (Γ) = 1 , where as usual | V (Γ) | is the number of vertices, | E (Γ) | is the number of internal edges, and ℓ (Γ) is thenumber of loops or independent cycles in Γ.Note that the external legs do not alter this relation. We have two types of edges, photons andfermions, so let us assume that p is the number of internal photon edges and f is the number of internalfermion edges, thus | E (Γ) | = p + f . Now, the most useful observation is that, in our case, we shouldhave p = ℓ (Γ). Indeed, we have seen that f = | V (Γ) | − , from which it follows that p = ℓ (Γ). This proves Claim 1.
So, we can generally think of graphs in Q as in the figure below, where P is the unique path formedby all directed fermion edges. P goes from f to f and passes through the vertex at r . All vertices lieon P . rest of the graph f f r This means we can uniquely put any graph Γ ∈ Q in the form of a rooted chord diagram, namelyby straightening P . See Figure 13 for an example. Figure 13.
A primitive quenched QED graph and its representation as a chord diagramFor simplicity of drawing we shall now and forth in the proof use dashed or light lines for photonsand drop the direction on the fermion edges on P . Also, let us agree that, in the chord diagramrepresentation, we will bring r to the front to play the role of a root, and still carry the information forthe external leg position at its other end. Thus, for example, the graph in Figure 13 is now representedas follows: The chord diagram representation of Γ ∈ Q will be denoted by C (Γ). Let us also denote the right endof the root r by v . The only property of Q that we still haven’t used is that a graph in Q is primitive. Claim 2:
A 1PI quenched QED graph Γ is primitive if and only if it is 2-connected in the chorddiagram representation. Subdivergences are translated into either a disconnection or a bridge.
Case 1:
Assume that C (Γ) is disconnected. This means that there exists an isolated componentof chords to the right or left of v . On the original graph this is simply a propagator-type subdivergenceinserted on one of the fermion edges. For an example see Figure 14 below. The converse is also clearly Figure 14.
Disconnections and propagator subdivergences.true, a propagator subdivergence is translated into an isolated component in C (Γ). Case 2:
Assume that C (Γ) has a reason S for connectivity-1, in the sense of Definition 2.7. Thenthe cut c for S is either the root chord r or not.(A) If c = r , then in Γ, S together with r correspond to a vertex-type subdivergence inserted at thevertex of the photon edge r .(B) If c = r , then S lies to the right or left of v in C (Γ). On Γ, this is a vertex-type subdivergenceinserted an end of the photon edge c .Conversely, by the same means, every vertex-type subdivergence in Γ gives a reason for connectivity-1 in C (Γ). This proves Claim 2 . Figure 15 illustrates situations (A) and (B) on one and the samegraph.Thus, every graph in Q is uniquely represented as a 2-connected chord diagram with the number ofchords equal to the number of internal photon edges and also equal to the loop number of the graph.The generating series z | ψ c | ( ~ ren ) counts the same diagrams, it only differs in not having the externalphoton leg r . The removal of the external leg r will not change the argument above: propagator-typesubdivergences correspond to isolated components in the chord diagram representation and vertex-typesubdivergences correspond to reasons for connectivity-1. This completes the proof. (cid:3) N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 21 r v c r c vertex-type subdivergences v reasons for connectivity-1 Figure 15.
Connectivity-1 and vertex-type subdivergences.This gives A z φ c | ψ c | ( ~ ren ) = ~ ren A z | ψ c | ( ~ ren ) = A C > ( x ) , and by equation (4.4):[ ~ n − ] z φ c | ψ c | ( ~ ren ) = [ ~ n ren ] z | ψ c | ( ~ ren ) = [ x n ] C > ( x ) == e − (cid:18) (2 n − − n − − n − − n − −− n − − n − − · · · (cid:19) = e − (2 n − (cid:18) − n − − n − n − − n − n − n − −− n − n − n − n − − n − · · · (2 n − − · · · (cid:19) , (6.1)which coincides with the result in [ ]. Appendix A. Factorially Divergent Power Series
This section aims to provide the necessary background for factorially divergent power series , asintroduced in Chapter 4 in [ ]. In [
5, 3 ], M. Borinsky studied sequences a n whose asymptotic behaviourfor large n follows a relation like(A.1) a n = α n + β Γ( n + β ) (cid:18) c + c α ( n + β −
1) + c α ( n + β − n + β −
2) + · · · (cid:19) , where α ∈ R > , and β, c k ∈ R . We will need to use the usual big and small o-notation for asymptoticanalysis: Given a sequence a n , O ( a n ) will denote the class of sequences b n satisfying lim sup n →∞ | b n a n | < ∞ ; whereas o ( a n ) shall denote the sequences b n such that lim n →∞ b n a n = 0. Moreover, a n = b n + O ( c n )should mean that a n − b n ∈ O ( c n ). Following [ ], we adopt the notation Γ αβ ( n ) := α n + β Γ( n + β ), whereΓ( z ) = R ∞ x z − e − x dx for Re( z ) > Definition
A.1 (Factorially Divergent Power Series) . For real numbers α and β , with α >
0, thesubset R [[ x ]] αβ of R [[ x ]] will denote the set of all formal power series f for which there exists a sequence( c fk ) k ∈ N of real numbers such that(A.2) f n = R − X k =0 c fk Γ αβ ( n − k ) + O (Γ αβ ( n − R )) , for all R ∈ N (positive integers). Remark
A.1 . From the definition it follows that R [[ x ]] αβ is a linear subspace of R [[ x ]]. Remark
A.2 . Also, by the above definition all real power series with a non-vanishing radius ofconvergence belong to R [[ x ]] αβ , with c fk = 0 for all k since in this case f n = o (Γ αβ ( n − R )) for all R ∈ N .The following proposition also follows directly from the definition. Proposition
A.1 ([ ], Ch.4) . The sequence ( c fk ) k ∈ N is unique for every f ∈ R [[ x ]] αβ ; actually c fN = lim n →∞ f n − P N − k =0 c fk Γ αβ ( n − k )(Γ αβ ( n − N )) for N ∈ N . Proposition
A.2 ([ ], Prop 4.3.1) . Given α, β ∈ R , with α > , the set R [[ x ]] αβ is a subring of R [[ x ]] . Note that the identity in (A.2) stands for an asymptotic expansion with asymptotic scale α n + β Γ( n + β ) (refer to [ ] for a detailed literature on the topic). The ring R [[ x ]] αβ is referred to as a ring offactorially divergent power series . Now, given f ∈ R [[ x ]] αβ , we can associate the coefficients ( c fk ) k ∈ N ofthe asymptotic expansion with a new ordinary power series: Definition
A.2 ([ ]) . For α, β ∈ R , with α >
0, let A αβ : R [[ x ]] αβ → R [[ x ]] be the map that has thefollowing action for every f ∈ R [[ x ]] αβ ( A αβ f )( x ) = ∞ X k =0 c fk x k . This interpretation allows us to use the coefficients c fk to get various results as we will see later. In[ ] M. Borinsky provides an extensive analysis for the map A αβ , we will include some of the propertieswithout proof and will try to include proofs only as much as needed. Remark
A.3 . A map of this type is called an alien derivative (operator) in the context of resurgencetheory [ ]. We will use this terminology occasionally. Remark
A.4 . Form the definition we see that A αβ is linear. A αβ is not injective since it vanishes forpower series with nonzero radius of convergence as mentioned above in Remark A.2. Proposition
A.3 ([ ], Prop 4.1.1) . For m ∈ N , f ∈ R [[ x ]] αβ if and only if f ∈ R [[ x ]] αβ + m and A β + m f ∈ x m R [[ x ]] . In this case x m (cid:0) A αβ f (cid:1) ( x ) = (cid:0) A αβ + m f (cid:1) ( x ) . Proof.
First note thatΓ αβ ( n ) = α n − m + β + m Γ( n − m + β + m ) = Γ αβ + m ( n − m ) . Now, f n = R − P k =0 c fk Γ αβ ( n − k ) + O (Γ αβ ( n − R )) , for all R ∈ N , can be re-indexed as f n = R ′ − X k = m c fk − m Γ αβ ( n − k + m ) + O (Γ αβ + m ( n − R ′ )) , for all R ′ > m. By the observation at the beginning the latter is equivalent to f n = R ′ − X k = m c fk − m Γ αβ + m ( n − k ) + O (Γ αβ + m ( n − R ′ )) , for all R ′ > m. Which proves the first part of the statement. Also, in that case (cid:0) A αβ + m f (cid:1) ( x ) = P ∞ k = m c fk − m x k = x m (cid:0) A αβ f (cid:1) ( x ) ∈ x m R [[ x ]]. (cid:3) The following corollary now follows.
Corollary
A.4 . For all m ∈ N , R [[ x ]] αβ ⊂ R [[ x ]] αβ + m . Thus we can always assume that β >
0, which is very convenient in deriving many results for thering R [[ x ]] αβ [ ]. N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 23
Proposition
A.5 ([ ], Prop 4.1.2) . For m ∈ N , f ∈ R [[ x ]] αβ ∩ x m R [[ x ]] if and only if f ( x ) x m ∈ R [[ x ]] αβ + m . In this case (cid:0) A αβ f (cid:1) ( x ) = (cid:0) A αβ + m f ( x ) x m (cid:1) ( x ) . Proof.
Again, since Γ αβ ( n + m ) = α n + β + m Γ( n + β + m ) = Γ αβ + m ( n ) , we can argue as follows:The ‘only if part’ follows by Proposition A.3. For the ‘if’ part, assume that g ( x ) = f ( x ) x m ∈ R [[ x ]] αβ + m .This gives that f n + m = g n = R − X k =0 c gk Γ αβ + m ( n − k ) + O (Γ αβ + m ( n − R )) , for all R ∈ N . By the observation above this is equivalent to f n + m = R − X k =0 c gk Γ αβ ( n + m − k ) + O (Γ αβ ( n + m − R )) , for all R ∈ N . Thus, for all n > m , f n = R − X k =0 c gk Γ αβ ( n − k ) + O (Γ αβ ( n − R )) , for all R ∈ N , which gives the desired result. Also, from the equations above we see that (cid:0) A αβ f (cid:1) ( x ) = (cid:0) A αβ + m f ( x ) x m (cid:1) ( x ). (cid:3) The next two theorems will be used later in the thesis, the proofs however are lengthy and requiremany lemmas, and shall thereby be omitted. The reader can refer to [ ] for the complete treatment. Theorem
A.6 ([ ], Prop 4.3.1) . Let α, β ∈ R , with α > . The linear map A αβ is a derivation overthe ring R [[ x ]] αβ , that is ( A αβ ( f · g ))( x ) = f ( x )( A αβ g )( x ) + g ( x )( A αβ f )( x ) , for all f, g ∈ R [[ x ]] αβ . More interestingly, the next theorem serves as a powerful tool for our purposes. For notation, weset Diff id ( R ,
0) = ( { g ∈ R [[ x ]] : g = 0 , g = 1 } , ◦ ), the group of formal diffeomorphisms tangent to theidentity, under composition of maps. Similarly, we set Diff id ( R , αβ = ( { g ∈ R [[ x ]] αβ : g = 0 , g = 1 } , ◦ )(easily checked to be a monoid). Theorem
A.7 ([ ], Th. 4.4.2) . Let α, β ∈ R , with α > . Then Diff id ( R , αβ is a subgroup of Diff id ( R , ; moreover, for any f ∈ R [[ x ]] αβ and g ∈ Diff id ( R , αβ the following statements hold: (1) f ◦ g and g − are again elements in R [[ x ]] αβ . (2) The derivation A αβ satisfies a chain rule, namely (A.3) ( A αβ ( f ◦ g ))( x ) = f ′ ( g ( x ))( A αβ g )( x ) + (cid:18) xg ( x ) (cid:19) β e g ( x ) − xαxg ( x ) ( A αβ f )( g ( x )) , and (A.4) ( A αβ g − )( x ) = − ( g − ) ′ ( x ) (cid:18) xg − ( x ) (cid:19) β e g − x ) − xαxg − x ) ( A αβ g )( g − ( x )) . It is worth mentioning here that this theorem offers more flexibility than the result by E. Benderin [ ]. Appendix B. The Hopf Algebra of Feynman Diagrams
This section is a quick review of the algebraic treatment of renormalization in terms of Hopf al-gebras. The definitions in this section will be needed to give sense of some of the expressions thatour results are related to. We have seen what renormalization is about analytically in the overviewof the work by Bogoliubov, Parasiuk, Hepp, and Zimmermann around 1960’s. Almost five decadeslater, D. Kreimer [ ] showed that the BPHZ scheme is captured by Hopf algebras and the recursivedefinition of the antipode. In this approach, the Feynman diagrams are used to define a connectedgraded and commutative Hopf algebra. For an extensive treatment of Hopf algebras see [ ], and see[
13, 9, 21, 22, 27, 19 ] for a spectrum of results and developments of the approach in renormalizationin QFT.
B.1. Basic Definitions of Hopf Algebras.
We let K be an infinite field (characteristic 0). Theunit of an algebra will be treated as a map in the sense of category theory. L ( V, W ) will mean the groupof K -linear maps from the K -vector space V to the K -vector space W . Definition
B.1 (Associative unital algebra) . An associative unital K -algebra ( A, m, I ) is a K -vector space A together with two linear maps m : A ⊗ A → A (product) and I : K → A (unit) such that: m ◦ (id ⊗ m ) = m ◦ ( m ⊗ id) m ◦ ( I ⊗ id) = m ◦ (id ⊗ I ) . Furthermore, the algebra is said to be commutative if m = m ◦ τ , where τ is the twist map a ⊗ b b ⊗ a .The image I (1) of 1 under the unit map I will often be denoted also by I with no confusion.In terms of commutative diagrams, A is an associative unital algebra if the following diagramscommute A ⊗ A ⊗ A id ⊗ m (cid:15) (cid:15) m ⊗ id / / A ⊗ A m (cid:15) (cid:15) A ⊗ A m / / A K ⊗ A ∼ = ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ I ⊗ id / / A ⊗ A m (cid:15) (cid:15) A ⊗ K id ⊗ I o o ∼ = u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ A The categorical dual then becomes
Definition
B.2 (Coassociative counital coalgebra) . A coassociative counital K -coalgebra ( C, ∆ , ˆ I )is a K -vector space C together with two linear maps ∆ : C → C ⊗ C (coproduct) and ˆ I : C → K (counit)such that : (id ⊗ ∆) ◦ ∆ = (∆ ⊗ id) ◦ ∆(ˆ I ⊗ id) ◦ ∆ = (id ⊗ ˆ I ) ◦ ∆ . Furthermore, the coalgebra is said to be cocommutative if ∆ = τ ◦ ∆.In terms of commutative diagrams, A is an associative unital algebra if the following diagramscommute C ⊗ C ⊗ C C ⊗ C ∆ ⊗ id o o C ⊗ C id ⊗ ∆ O O C ∆ o o ∆ O O K ⊗ C C ⊗ C ˆ I ⊗ id o o id ⊗ ˆ I / / C ⊗ K C ∼ = ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ ∆ O O ∼ = i i ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ Sweedler’s notation for the coproduct is often useful: ∆( x ) = P x x ′ ⊗ x ′′ . Definition
B.3 (Algebra morphism) . Let (
A, m A , I A ) and ( B, m B , I B ) be two associative unital K -algebras. A K -linear map ϕ : A −→ B is an algebra morphism if ϕ ◦ I A = I B , and ϕ ◦ m A = m B ◦ ( ϕ ⊗ ϕ ) . N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 25
Dually, one defines
Definition
B.4 (Coalgebra morphism) . Let ( C, ∆ C , ˆ I C ) and ( D, ∆ D , ˆ I D ) be two coassociativecounital K -coalgebras. A K -linear map ψ : C −→ D is a coalgebra morphism ifˆ I D ◦ ψ = ˆ I C , and∆ D ◦ ψ = ( ψ ⊗ ψ ) ◦ ∆ C . Definition
B.5 (Bialgebras) . A K -bialgebra ( B, m, I , ∆ , ˆ I ) is a K -vector space such that ( B, m, I )is an algebra and ( B, ∆ , ˆ I ) is a coalgebra and that the two structures are compatible in the sense that m (and I ) is a coalgebra morphism and ∆ (and ˆ I ) is an algebra morphism.Note that only one of the compatibility conditions is enough; one can verify that m is a coalgebramorphism if and only if ∆ is an algebra morphism.Note that in a bialgebra it must be that ˆ I ( I ) = 1 and vanishes for all other elements. Definition
B.6 (Hopf algebras and the antipode) . A Hopf algebra ( H , m, I , ∆ , ˆ I , S ) is a K -bialgebra( H , m, I , ∆ , ˆ I ) together with a linear map S : H → H such that m ◦ ( S ⊗ id) ◦ ∆ = I ◦ ˆ I = m ◦ (id ⊗ S ) ◦ ∆ . The map S is called the antipode of the Hopf algebra.Diagrammatically this is equivalent to the following diagram being commutative: H ⊗ H S ⊗ id / / H ⊗ H m ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗ H ∆ (cid:15) (cid:15) ∆ O O ˆ I / / K I / / HH ⊗ H id ⊗ S / / H ⊗ H m ♠♠♠♠♠♠♠♠♠♠♠♠♠♠ Definition
B.7 (Convolution Product) . Let f, g be two linear maps in L ( H , H ). Then their convolution product is defined as f ∗ g := m ◦ ( f ⊗ g ) ◦ ∆ . This product gives again a linear map on H . It can be shown that ( L ( H , H ) , ∗ , I ◦ ˆ I ) becomes analgebra. Moreover, f ◦ S is the inverse of f with respect to the convolution product, and in that sense,the antipode S may be thought of as the ∗ -inverse of the identity map id H .B.1.1. Filtration and Connectedness of Hopf Algebras.
Definition
B.8 (Gradedness and Connectedness) . A Hopf algebra H is said to be graded ( Z > -graded to be precise) if it decomposes into a direct sum H = ⊕ ∞ n =0 H n , such that m ( H n ⊗ H m ) ⊆ H n + m , ∆( H n )) ⊆ ⊕ nk =0 H k ⊗ H n − k ,S ( H n ) ⊆ H n . If, in addition, H ∼ = K , the Hopf algebra is said to be connected .Given a graded Hopf algebra as above, one finds that ker ˆ I = Aug H := ⊕ ∞ n > H n , called the augmentation ideal . Definition
B.9 (Filtration) . A Hopf algebra H is filtered if there exists a tower of subspaces H n ⊆ H n +1 , n ∈ N , such that H = ∞ X n =0 H n ,m ( H n ⊗ H m ) ⊆ H n + m , ∆( H n )) ⊆ n X k =0 H k ⊗ H n − k ,S ( H n ) ⊆ H n . Note that every graduation implies a filtration by taking H n = ⊕ nk =0 H k . Definition
B.10 (Primitive and group-like elements) . An element x ∈ H is primitive if ∆( x ) = I ⊗ x + x ⊗ I . An element x is group-like if ∆( x ) = x ⊗ x . B.2. Physical Theories as Combinatorial Classes of Graphs.
Now we are ready to definethe renormalization Hopf algebras of Feynman diagrams, but first let us emphasize the combinatorialrephrasing of the physical setup already seen in the previous sections.All of the enumerative aspects of QFT considered in this thesis are about Feynman diagrams.Feynman diagrams and their Hopf algebras will be key ingredients in the later chapters, although notexplicitly affecting the combinatorial problems we consider. We will proceed by defining combinatorialQFT theories, Feynman graphs, and Feynman rules. Then we will recover some of the concepts ofrenormalization. Recommended references for similar treatments are [
33, 27, 5 ].The building block for Feynman graphs is going to be half edges . An edge is intuitively understoodto be formed from two half edges.
Definition
B.11 . A graph (or diagram) G is a set of half edges for which there is(1) a partition V ( G ) into disjoint classes of half edges, a class v ∈ V ( G ) will be called a vertex ;(2) a collection E ( G ) of disjoint pairs of half edges. E ( G ) will be called the set of internal edges ;(3) half edges that are not occurring in any of the pairs in E ( G ) will be called external edges or external legs .The size of a graph will be the size of its set of half edges. Half edges can be labelled or unlabelled,and sometimes we will use many types of half edges to represent a certain physical theory. We will beconcerned at some point with graphs which have a prescribed set of external legs. The loop number ofa graph is the dimension of its cycle space, or in other words the number of independent cycles. Thereexists, in any graph, a family of independent cycles with each cycle having an edge not occurring in anyof the other cycles in the family. The size of the largest such family is the number of independent cyclesin the graph. Such a family of cycles can be obtained by starting with a spanning tree and readingoff the new cycle created by adding one of the edges, one edge at a time (and with removing any edgeadded earlier). The loop number will be very important in our later considerations and will expresssome sort of size for diagrams with a prescribed scheme of external legs.By Aut( G ) we mean the group of automorphisms (self isomorphisms) of the graph G . In theperturbative expansions that we will see, a Feynman diagram will have a symmetry factor of 1 / | Aut( G ) | (it is more common to write it as 1 / Sym( G ). We will usually work with unlabelled graphs, nevertheless,the symmetry factor will allow us to use the exponential relation between connected and disconnectedobjects.The good thing about many of the aspects of quantum field theory is that they can be transformedinto purely combinatorial and enumerative problems. Definition
B.12 . A combinatorial physical theory consists of(1) a dimension of spacetime (nonnegative integer);(2) a number of half edge types, and a set of pairs of half edge types, with each pair representing anadmissible edge type in the theory (note that the half edge types in one pair are not necessarilydistinct nor identical);(3) a collection of multisets of half edge types to define the options for a vertex in the theory; N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 27 (4) an integer weight for each edge or vertex type, called a power counting weight .Thus, a graph in a certain theory T will be a graph whose edges are of the types formed by theadmissible pairs of T , and each of whose vertices is incident to an admissible multiset of half edges.Note that even oriented and unoriented edges can be formed this way: oriented edges arise from anadmissible pair of half edges in which the two types are different, whereas unoriented ones arise frompairs with the two types identical. Example
B.1 . (1) QED: Quantum electrodynamics.
In QED there are 3 half edge types:a half photon, a front half fermion, and a back half fermion. The admissible combinations ofhalf edges to form edges are: (1) a pair of two half photons to give a photon edge, drawn as awiggly line , with power counting weight 2; and (2) a pair consisting of a frontand back halves fermion to give a directed fermion edge , with power countingweight 1. There is one type of a vertex, namely, a vertex is 3-valent and is incident to one ofeach half edge type, with power counting weight 0. The spacetime dimension is taken to be 4.(2)
Yukawa theory:
This theory also has 3 types of half edges: a half meson, a front half fermion,and a back half fermion. The admissible edges are: (1) a meson edge formed by two half mesons(front and back), drawn as , and has power counting weight 2; and (2) a pairof a front and back halves fermion to give a directed fermion edge , with weight 1.Just as in QED, there is one type of vertices, namely, a vertex is 3-valent and is incident toone of each half edge type, with power counting weight 0. The spacetime dimension is takento be 4. The difference from QED lies in the Feynman rules.As mentioned earlier, the significance of quantum field theory is the ability to describe how particlesinteract and scatter. In an idealized experiment some particles are sent in, they interact and scatter, andthen the outcomes are detected. This picture can be visualized as a diagram in which the edges describepropagating particles. The idea then is that, on an atomic scale, we never know what exactly happenedand every possible interaction is assigned a probability scattering amplitude . This amounts into aweighted sum, known as a perturbative expansion . The probabilities in the theory are computed throughwhat is known as a
Feynman integral . These integrals encountered by physicists are often divergentand have to undergo renormalization to retrieve useful information. As we saw before, Feynman graphsencode these complicated integrals, and the rules for this encoding in a given QFT are known as
Feynmanrules . Definition
B.13 (Feynman Graphs) . A Feynman graph in a theory T is combinatorially a graphstructure in which edges fall into certain types and vertices are subject to conditions (pertinent to T ) onthe number of edges of a certain types attached to it. A Feynman graph represents an integral throughthe Feynman rules of the theory, which assigns an integrand factor contribution to every internal edgeor vertex. The power counting weights give the degree of an integrated momentum variable. Example
B.2 . In the following example (Figure 16) the integral is a divergent Feynman integraland its corresponding Feynman diagram: k k − ℓ ℓ − q ℓ + p qp + qk + pp = Z Z d D ℓℓ ( ℓ − q ) ( ℓ + p ) d D kk ( k − ℓ ) ( k + p ) . Figure 16.
The Feynman integral of a Feynman graph
B.3. Characters and Cocycles.
Definition
B.14 (Characters) . Let H be a connected bialgebra and ( A, · , A ) be a K -algebra. A character from H to A is defined to be an algebra morphism with the extra property that φ ( I ) = 1 A .The set of all characters from H to A is denoted by G H A . Further, if A is commutative, G H A becomes agroup under convolution product [ ]. The inverses are denoted ϕ ∗− := ϕ ◦ S . Definition
B.15 . Let H be a connected bialgebra and A be a commutative algebra that can bewritten as a direct sum of two vector spaces. A Birkhoff decomposition of a character φ is a pair ofcharacters φ + , φ − ∈ G H A such that φ = φ ∗− − ∗ φ + and φ ± (kerˆ I ) ⊆ A ± . In [ ] it was shown that dimensional regularization (viewing the integral over dimension D − ǫ andexpanding in ǫ ) can be studied in terms of characters into the algebra of meromorphic functions in ǫ . Theorem
B.1 ([ ]) . Let H be a connected filtered Hopf algebra, and let G H A be the group of char-acters with the convolution product. Then any character ϕ ∈ G H A has a unique Birkhoff decomposition ϕ = ϕ ∗− − ∗ ϕ + , where ϕ − , ϕ + ∈ G H A , with ϕ − mapping the augmentation ideal into A − , and with ϕ + mapping H into A + . The characters naturally satisfy ϕ − ( I ) = 1 A = ϕ + ( I ) and are defined recursively over theaugmentation ideal as ϕ − ( x ) = − π (cid:0) ϕ ( x ) + X x ϕ − ( x ′ ) ϕ ( x ′′ ) (cid:1) , and ϕ + ( x ) = (id − π ) (cid:0) ϕ ( x ) + X x ϕ − ( x ′ ) ϕ ( x ′′ ) (cid:1) , where π is the projection of A onto A − , and the sum is making use of Sweedler’s notation for thecoproduct. Definition
B.16 (Bogoliubov map) . The
Bogoliubov map is the map b : G −→ Hom( H , A ) definedrecursively by b ( ϕ )( x ) = ϕ ( x ) + X x ϕ − ( x ′ ) ϕ ( x ′′ ) . In particular, the decomposition in Theorem B.1 is now seen via the Bogoliubov map as ϕ − = − π ◦ b ( ϕ ) , and ϕ + = (id − π ) ◦ b ( ϕ ) . Before starting the next part, it must be noted that this section does not give a full account of theHopf-algebraic treatment of renormalization. We are only interested in defining the expressions that wewill encounter in our problems. The reader can refer to [ ] for an in depth account on Hopf algebras.The Hopf algebra of Feynman graphs is also surveyed in the review article of D. Manchon [ ]. B.4. The Hopf Algebra of divergent 1PI Diagrams.
Let T be a fixed combinatorial physicaltheory in the sense of the previous section, and consider the Q -vector space H generated by the set ofdisjoint unions of divergent 1 P I
Feynman graphs in the theory T , including the empty graph which wedenote by I .We can define a multiplication m on H to be taking the disjoint union, and the unit is the emptygraph I , this makes ( H , m, I ) a commutative associative algebra.Now we need to define a compatible coalgebra structure for H .In the next just note that the residue of a Feynman graph is simply the graph obtained if all internaledges were contracted. For the sake of a precise general definition of contraction in the new terms wehave Definition
B.17 (Contraction of a Subgraph) . Let Γ be a Feynman graph in a theory T , and let γ ⊆ Γ be a subgraph each of whose connected components is 1
P I and divergent. The contraction graph Γ /γ is constructed as follows: N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 29 (1) A component of γ with a vertex residue (external leg structure) is contracted in Γ into a vertexof the same type as the residue.(2) A component of γ with an edge residue (external leg structure) is contracted in Γ into an edgeof the same type as the residue.The superficial degree of divergence ω (Γ) of a graph Γ is defined in many references ([
33, 19, 26 ])in which it is also shown that ω (Γ) = D ℓ (Γ) − P a w ( a ), where the sum is over all the power countingweights (Definition B.12) of vertices and internal edges in Γ determined by the QFT theory considered,and where ℓ (Γ) is the number of loops in Γ. Definition
B.18 (Subdivergence) . A subgraph γ of Γ with divergent 1 P I connected componentsis called a subdivergence .Then we define the coproduct as
Definition
B.19 . The coproduct ∆ :
H −→ H ⊗ H is defined for a connected Feynman graph Γto be ∆(Γ) = X γ ⊆ Γ γ product of divergent1 P I subgraphs γ ⊗ Γ /γ and extended as an algebra morphism.Note that since we are considering graphs that are themselves divergent, the coproduct sum for anyelement in H will always start as ∆(Γ) = I ⊗ Γ + Γ ⊗ I + e ∆(Γ) . The part e ∆(Γ) of the coproduct is called the reduced coproduct . Definition
B.20 (Primitive Elements) . An element Γ ∈ H is said to be primitive if e ∆(Γ) = 0.That is, ∆(Γ) = I ⊗ Γ + Γ ⊗ I . In particular, a primitive 1 P I graph Γ is a 1
P I graph that contains nosubdivergences in the sense of Definition B.18.For example let us calculate the coproduct∆ (cid:18) − (cid:19) = I ⊗ (cid:18) − (cid:19) + (cid:18) − (cid:19) ⊗ I ++ ⊗ − ⊗ = I ⊗ (cid:18) − (cid:19) + (cid:18) − (cid:19) ⊗ I − ⊗ . Finally, let ˆ I : H −→ Q be the map defined on the empty graph by sending q I to q ∈ Q andsending every other element in H to zero. Then it is not hard to prove the following proposition (see[
33, 2, 32 ] for a proof)
Proposition
B.2 . As per the above definitions, ( H , m, I , ∆ , ˆ I ) is a bialgebra. Further, if we definea map S : H −→ H recursively by S ( I ) = I , and S (Γ) = − Γ − X γ ⊆ Γ I = γ =Γ γ product of divergent P I subgraphs S ( γ ) Γ /γ, then ( H , m, I , ∆ , ˆ I , S ) becomes a Hopf algebra woth antipode S . (Note that the product in the secondterm is the product m abbreviated). Moreover, the Hopf algebra H is commutative and is graded by theloop number. Remark
B.1 . In the next section we will broadly see how renormalization is represented in thisalgebraic context of Hopf algebras. Our job ends with learning the meaning of some of the expressionsthat will show up again in our problems. It should be noted however that, as expected, this is notthe only meaningful appearance of Hopf algebras in quantum field theory. Namely, if the conditionof divergence is dropped from the elements summed over in the definitions of the coproduct and theantipode, we get the so-called the core Hopf algebra , denoted H c . It turns out that H c interplays withCutkosky cuts in graphs, this is related to the unitarity of the S -matrix [ ]. B.5. Renormalization in Hopf algebras.
B.5.1.
Feynman Rules and Characters of H : Let a theory T be fixed as before, and let H be theHopf algebra generated by sets of divergent 1 P I
Feynman graphs in T . We start by thinking of Feynmanrules as a map φ that assigns formal integrals to elements in H , and we investigate what conditionsshould be imposed on φ to fully interpret the Feynman rules.For the Feynman rules, we need to satisfy certain criteria:(1) The map φ should be multiplicative on disjoint unions of graphs. Moreover, the map shouldalso have a multiplicative property for bridges. The latter requirement enables us to startdefining φ over 1 P I diagrams. The leap from all Feynman graphs to 1
P I graphs is donethrough the Legendre transform, which has been redefined recently as a purely combinatorialmap [
18, 17 ].(2) The map φ , representing Feynman rules, has also to adapt with the combinatorial Dyson-Schwinger equations. Precisely, it has to interplay nicely with the process of insertion whichwe discuss in the next section. (A) All of this was seen to suggest that the Feynman rules are to be represented by a character φ ∈ G H A , where A is a suitably chosen commutative algebra. The target algebra A is usually taken tobe the algebra C [ L ][[ z − , z ]] of Laurent series whose coefficients are polynomials in an energy scale L .For example, in [ ], L = log( q /µ ) where q and µ are the external momenta and the renormalizationscale respectively.B.5.2. Rota-Baxter Operators:
Let A be an algebra as before. An operator (linear map on A ) R : A −→ A is said to be a Rota-Baxter operator if it satisfies R [ ab ] + R [ a ] R [ b ] = R (cid:2) R [ a ] b + a R [ b ] (cid:3) , for all a, b ∈ A .Indeed, it turns out that the truncated Taylor operator T ω (Γ) in the BPHZ scheme is a Rota-Baxteroperator. This relation between renormalization and Rota-Baxter operators has been extensively studiedin [
13, 12 ]. (B) In general, a Rota-Baxter operator will be used to express a map which sends a formal integralto the evaluation of the integral at the subtraction point in the renormalization scheme. In other words, R produces the counterterms. If Γ is a divergent graph with no subdivergences, Rφ (Γ) will stand forthe ill part of the integral φ (Γ).It remains to setup a technology for dealing with subdivergences recursively.Define a linear map S φR : H −→ A by S φR ( I ) = 1 A and N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 31 (B.1) S φR (Γ) = − R ( φ (Γ)) − X I = γ ( Γ γ product of divergent1 P I subgraphs S φR ( γ ) R ( φ (Γ /γ )) , and extended to all of H as a morphism of algebras. (C) Then the renormalized
Feynman rules are defined to be(B.2) φ R = S φR ∗ φ. It can be shown that φ R ( x ) = ( S φR ∗ φ )( x ) = (id A − R ) b ( φ )( x ) , where b is the Bogoliubov mapdefined before (Definition B.16) [ ].By (A), (B), and (C), the conclusion is that the approach of renormalization is as follows: (1) Weexpress Feynman graphs in a graded Hopf algebra H , and interpret the Feynman rules as charactersfrom H to some commutative algebra A . (2) A renormalization scheme is determined via a Rota-Baxteroperator on A , this also determines a Birkhoff decomposition A = A − ⊕ A + into two subalgebras. (3)The renormalized Feynman rules are obtained through the coproduct and the map S φR . For explicitexamples and applications of this approach the reader can refer to [
19, 33, 27 ]. B.6. Combinatorics of Dyson-Schwinger Equations.
B.6.1.
Insertions.
Definition B.17 introduces the notion of contracting a subgraph within a biggergraph. One can think of a reverse operation in terms of inserting a graph γ into another graph Γ as asubgraph, in one of the potential positions ( insertion places ) in Γ that can host γ . An insertion placehas to be compatible with the external leg structure of the graph being inserted. Definition
B.21 (Insertion) . Let γ be a Feynman graph with external leg structure r = res( γ ).Let Γ be a Feynman graph with a vertex or an internal edge of the same type as r .(1) If r is of edge type, and e is an internal edge in Γ of the same type, then we can insert γ intoΓ as follows:Break the edge e into two half edges, each of which is identified with one of the twocompatible external legs of γ .(2) If r is of vertex type, and v is a vertex of the same type in Γ, then we can insert γ into Γ asfollows: Break every edge incident to v , and, in a compatible way, which may not be unique,attach the external legs of r to the resulting half edges in Γ − v .The places e or v in the above scenarios are called insertion places . Notice that the way to insert γ into Γ at a certain insertion place is not unique and depends on the symmetries of the graphs.We wish now to define an operator B Γ+ that inserts graphs into a fixed graph Γ. Definition
B.22 ([ ]) . For a connected 1
P I
Feynman graph Γ we define B Γ+ ( X ) = X G bij(Γ , X, G ) | X | ∗ G ) 1(Γ | G ) G, where(1) maxf( G ) is the number of insertion trees corresponding to G ,(2) | X | ∗ is the number of distinct graphs obtained from permuting the external legs in X ,(3) bij(Γ , X, G ) is the number of bijections of the external legs of X which have an insertion placein Γ so that the insertion gives G .(4) (Γ | G ) is the number of insertion places for X in Γ. Remark
B.2 . See [ ] Theorem 4 for a justification of this definition. In the case of trees, theoperation B + ( T · · · T m ) takes the rooted trees T , · · · , T m and attach all of their roots as children of anew added root, getting a single rooted tree (the name grafting operator makes sense in this case). In the case of rooted trees described in the remark above, if H C denotes the Connes-Kreimer Hopfalgebra of rooted trees [
8, 33 ], then the grafting operator B + is characterized as being a Hochschild -cocycle , that is: ∆ ◦ B + ( t ) = (id ⊗ B + ) ◦ ∆( t ) + B + ( t ) ⊗ I . This property will be highlighted in the next section as it is crucial to the algebraic reconstructionof renormalization in the approach pioneered by D. Kreimer and his collaborators. For more about thisalgebraic treatment and concepts see the original paper by D. Kreimer and A. Connes [ ] or [ ]. Remark
B.3 . In order to get a 1-cocycle from the operators B γ + it turns out that we can not workwith individual primitive graphs, rather, we should sum over all primitive graphs of a given loop number[
33, 23 ]To express the Dyson-Schwinger equations in terms of the operators B Γ+ we need to know moreabout the number of insertion places in a given graph.Let us assume that the combinatorial theory we are considering now has only one vertex type v , andlet d be the degree of any such vertex. Also set n ( e ) to be the number of half edges of type e appearingin the external legs of vertex-type v . By definition we set n ( v ) = 1. Proposition
B.3 ([ ]) . Let Γ be a P I graph in a QFT theory of the type described above, thatis, the theory has only one vertex type v with d being the degree of such a vertex. Let r = res(Γ) , and ℓ = ℓ (Γ) (the loop number). Also let n ( e ) be the number of half edges of type e appearing in the externallegs of vertex-type v . Besides, define n ( v ) = 1 . Then (1) Γ has ℓn ( s ) d − insertion places for every type s = r ; (2) If r is vertex-type, then Γ has ℓn ( r ) d − insertion places for type r ; and (3) If r is not vertex-type, then Γ has − ℓn ( r ) d − insertion places for type r . Example
B.3 . In QED (quantum electrodynamics) we have only one vertex type, namely ,and two edge types: a photon edge , and a fermion edge .We are going to follow the notation used in [ ], namely • X vertex is the vertex series , whose n th coefficient is the sum of all 1 P I
QED diagrams withresidue and loop number n . • X photon is the photon edge series , whose n th coefficient is ( − ×{ the sum of all 1 P I
QEDdiagrams with residue and loop number n } . • X fermion is the fermion edge series , whose n th coefficient is ( − ×{ the sum of all 1 P I
QEDdiagrams with residue and loop number n } . Remark
B.4 . Notice that the negative signs with the edge series arise as we will be actuallyinterested in sequences of such diagrams, and so if Y is the original generating function then we are toget a geometric series 11 − Y . Then we use X = 1 − Y . N ASYMPTOTIC EXPANSION FOR THE NUMBER OF 2-CONNECTED CHORD DIAGRAMS 33
Then we have X vertex = I + X γ primitive withvertex residue x ℓ ( γ ) B γ + ( X vertex ) ℓ ( γ ) ( X photon ) ℓ ( γ ) ( X fermion ) ℓ ( γ ) ! , (B.3) X photon = I − x B + ( X vertex ) ( X fermion ) ! , (B.4) X fermion = I − x B + ( X vertex ) X photon X fermion ! . (B.5)These equations are obtained by a direct counting argument. For example, the second equation canbe illustrated through Figure17. Figure 17.
Graphs with photon edge residueIn Figure 17 the blue bubbles represent the two sequences of fermion-type 1
P I graphs that can beinserted along the original two fermion edges, hence the 1 / (cid:0) X fermion (cid:1) ; whereas the two larger greybubbles represent insertion of a vertex-type 1 P I graph and correspond to the ( X vertex ) . The minussign follows from the definition of X photon . B.7. The Invariant Charge.
If we set Q = ( X vertex ) ( X photon ) ( X fermion ) , then we can rewrite equations(B.3), (B.4), and (B.5) as X vertex = I + X γ primitive withvertex residue x ℓ ( γ ) B γ + (cid:16) X vertex Q ℓ ( γ ) (cid:17) , (B.6) X photon = I − x B + (cid:0) X photon Q (cid:1) , (B.7) X fermion = I − x B + (cid:0) X fermion Q (cid:1) . (B.8)The expression Q is to be called the invariant charge . In general, for a theory with only one vertextype v , the system of Dyson-Schwinger equations takes the form(B.9) X r = 1 ± X k B γ r,k + ( X r Q k ) , where the sum is over k and over all primitive 1PI diagrams γ r,k with loop number k and residue r .(B.10) Q = X v pQ e ∈ v ( X e ) ! d − , where d is the degree of the vertex-type, and the product is over all half edges making up the vertex.Note that the orientation of a half edge is ignored in counting the types. In some references, theconvention for the invariant charge is to be the square root of our definition [ ]. Also in [ ] the X r isdefined to be the negative of ours in case r is an edge type. References [1] E. Bender.
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Mathematics Department, Faculty of Science, Cairo University, Egypt
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