Triangle and box diagrams in coupled-channel systems from the chiral Lagrangian

We perform an analysis of triangle- and box-loop contributions to the generalized potential in the scattering of Goldstone bosons off the J^P= 0^- and 1^- charmed mesons. Particular emphasis is put on the use of on-shell mass parameters in such contributions in terms of a renormalization scheme that ensures the absence of power-counting violating terms. This is achieved with a systematically extended set of Passarino--Veltman basis functions, that leads to manifest power-counting conserving one-loop expressions and avoids the occurrence of superficial kinematical singularities. Compact expressions to chiral order three and four are presented that are particularly useful in coding such coupled-channel systems. Our formal results are generic and prepare analogous computations for other systems, like meson-baryon scattering from the chiral Lagrangian.


I. INTRODUCTION
The role of left-hand cut contributions in coupled-channel systems receives increasing attention in the hadron physics community as the interplay of modern effective field theory approaches with Lattice QCD simulations requests more and more quantitative and controlled computations. The open-charm sector of QCD not only serves as a convenient laboratory since it is largely driven by the symmetries of QCD [1-4], but also offers already a sizeable data set from Lattice QCD simulations [5][6][7][8][9][10][11][12][13]. We consider such studies as a preparation for the more demanding meson-baryon systems for which the scattering data set from Lattice QCD simulations is significantly more scarce [14][15][16][17][18][19][20][21][22][23][24][25].
Studies of the quark-mass dependence of the charmed meson masses are the key for the quantitative understanding of the coupled-channel interactions of the latter with the Goldstone bosons of QCD [26,27]. It is useful to acknowledge that simultaneous approaches for hadron masses together with their scattering properties are significantly more constrained by QCD as compared to partial studies. Early coupled-channel works in the open-charm sector focused on the s-wave interactions only and ignored the impact from the quark-mass dependence of the charmed meson masses [2,[28][29][30][31][32][33][34]. Coupled-channel studies of p-wave and d-wave systems are of equal importance, since in Lattice QCD studies or experimental cross section results, a focus on s-wave terms only is not always possible. For the latter the scattering processes cannot be reliably described by algebraic matrix equations (see e.g. [2,26,29]), that may lead to unitarity but are at odds with the long-range part of the coupled-channel forces as they arise from t-or u-channel exchange processes at the treeor loop-level [35][36][37][38]. We note that a suitable framework for such systems is offered by the generalized potential approach (GPA) as was developed in [39][40][41][42]. It systematically extends the applicability domain of the chiral Lagrangian into the resonance region by using an expansion of the generalized potential in terms of conformal variables, where the expansion coefficients are well accessible within Chiral Perturbation Theory (χPT).
In our current formal work we focus on one-loop triangle and box contributions that have not been studied at sufficient rigor from the chiral Lagrangian. While a first estimate of such effects was reported on in [43,44]  subleading chiral orders take the form (1) with Clebsch coefficients C W T , C We note that the particularly useful combination of Clebsch coefficients was introduced in [28] for applications in which the mass difference of the D * and D * s , or also the difference of the D and D s masses, can be neglected. Depending on the context such Clebsch coefficients are applied also for processes which involve the scattering of the This can be illustrated at the leading orders for the scattering processes involving the 1 − charmed mesons. The scattering amplitudes are characterized by six invariant amplitudes G n , most economically in the following choice (s, t, u) = G 0 (s, t, u) ϵ αβµνq α q β ϵ † µ (p,λ) p ν , (s, t, u) = ϵ † µ (p,λ) G 1 (s, t, u) g µν + G 2 (s, t, u)q µ q ν + G 3 (s, t, u)q µ (q −q) ν + G 4 (s, t, u) (q − q) µ q ν + G 5 (s, t, u) (q − q) µ (q −q) ν ϵ ν (p, λ) , where we usep µ andλ for the momentum and polarization of the produced D * meson. The wave function, ϵ αβ (p, λ) of a vector meson as interpolated by an antisymmetric tensor field, is written in terms of the more conventional wave function ϵ α (p, λ) of a spin-one particle in the vector-field representation. We identify the leading orders tree-level terms with where we use the notation Q n to specify the chiral order n of a given term. While theg 4 andg 5 contribute to the 0 − 1 − → 0 − 1 − processes, the heavy-quark symmetry related g 4 and g 5 contribute to the production processes 0 − 0 − → 0 − 1 − . In the heavy-quark mass limit it holdsg n = g n , in particular for n = 4, 5.
It remains to specify the invariant amplitude G 1 (s, t, u) in T Owing to the heavy-quark symmetry its form can be inferred from its spin-zero partner reaction T ab (s, t, u) at least in the heavy-quark mass limit. Indeed we find for our tree-level the expression properly truncated at chiral order four. This is so since contributions from the other amplitudes G 2,3,4,5 are suppressed by two orders in the chiral expansion. Note the presence of the small 4-momentaq µ or q µ in (3). It is evident that analogous relations hold for the loop expressions, as to be derived in our current work. Therefore from now on we focus on the reactions with spin-zero charmed mesons in the initial and final states.

III. SCATTERING WITH TADPOLE AND BUBBLE DIAGRAMS
We discuss one-loop contributions to the two-body scattering amplitudes. At chiral order 3 and 4 there are various types of diagrams to be considered. All one-loop diagrams that contribute at g P = 0 have been evaluated in our previous work [37]. Such tadpole and bubble loop contributions are recalled in Fig. 1 at order 3 involving leading order vertices only. Corresponding diagrams at order 4 involve subleading order vertices instead. Quite explicit expressions are documented in [37].
An additional set of tadpole, bubble, triangle and box loop diagrams is proportional to g 2 P has not been documented systematically before. In Fig. 2-7 our target diagrams are shown for the case that initial and final mesons carry J P = 0 − quantum numbers.
Corresponding diagrams can be drawn for the case in which one or both external lines signal a charmed meson with J P = 1 − . From the form of such diagrams it follows that in the formal limit of a very large mass of the J P = 1 − mesons such contributions may be viewed as a renormalization of tadpole and bubble loop contributions. That was the rationale behind our previous more phenomenological work, despite the fact that the heavyquark spin symmetry predicts the mass degeneracy of the 0 − and 1 − states in the limit of an infinite charm quark mass. Clearly, it is desirable to have a closer look into such diagrams.
We use the conventional Mandelstam variables s, t and u of two-body scattering. The where we use small m's for Goldstone boson masses and big M 's for the masses for the 0 − and 1 − charmed mesons. The pairs of initial and final four momenta are (q µ , p µ ) and (q µ ,p µ ) respectively. In turn we may write w = q + p =q +p with s = w 2 .
We first consider the tadpole-type diagrams in Fig. 2 with the wave-function factors of the Goldstone bosons as written in [37] by using the LEC L 4 and L 5 of Gasser and Leutwyler [50]. While the Clebsch C QH can easily be expressed in terms of the latter. To avoid a proliferation of our notations, Q is used as a placeholder index for a Goldstone boson field π, K, η in (7) but also in Q n together with the chiral order n of a given term (see e.g. (4)). For the tadpole,Ī Q , a conventional M S subtraction scheme is used with the renormalization scale µ of dimensional regularization.
We aim at a decomposition of the scattering amplitude H,ab as well as the background term B ab (s, u) receive corrections from loop effects. Given our approximation strategy we will use the physical on-shell mass for M H and G (s) from (1). The value g P may be adjusted as to recover the empirical decay width of the D * → π D meson. While it would be desirable to refine such a scheme, at this stage there is insufficient information available to consider flavor breaking or quark-mass dependence effects in G (s) H,ab . It appears impossible to determine corresponding LEC that contribute to G (s) H,ab . Therefore we will focus on the loop effects in the background term. For the tadpole contribution (7) we find where we note that the background term is of chiral order four. This is in contrast to its corresponding contribution to G H , which are of chiral order two. We turn to the bubble-type contributions, where we start with the wave-function term − g 2 from Fig. 3. It involves the first and second order tree-level expressions T (1) ab (s, u) and T (2) ab (s, u) as recalled in (1) and the LEC ζ 0 and ζ 1 from the chiral Lagrangian. Our result involves a scalar bubble loop functionĪ QR (M 2 H ), with its renormalized form given in (14). We find that the wave functions, Z H , of the heavy fields do not depend on the renormalization scale µ, if we use the summed expressions in (11). This is contrasted by the fact that the wave functions, Z Q , of the light fields do depend on µ. We note that an additional subtraction in (11) may be useful as to arrive at the wave function factors Z (2) D and Z (2) Ds to approach one in the chiral limit. The evaluation of the loop functions in Fig. 3-5 (see also (11), (17) and (21)) is straight-forward, even if one insists on the use of on-shell meson masses as is highly advisable for coupled-channel systems. In previous works [51][52][53][54] we developed a novel scheme in application of the Passarino-Veltman decomposition scheme [55]. In an initial step the one-loop where with M andM we denote the chiral limit values of the charmed meson masses with J P = 0 − and J P = 1 − respectively. Additional contributions from scalar tadpole integrals involving the heavy fields are dropped systematically withĪ L =Ī R =Ī H → 0, at least if they occur in a power-counting violating context. By construction it holdsĪ QH (s) ∼ Q as expected from dimensional counting rules. In a second step we apply the power-counting scheme [37] as introduced in terms of on-shell hadron masses  (15). An application of the counting rules (15) generates expressions that probe rational functions of that r, as is illustrated in [53,54].
There are at least three relevant possibilities implied by either r ∼ Q 2 or r ∼ Q or r ∼ Q 0 . In the first two cases we make contact with the traditional simultaneous expansion in the small up, down and strange quark masses and in the small inverse of a large charm quark mass.
In the third case we may integrate out the 1 − fields in terms of the formal requestM ≫ M .
Most economic would be the case with r ∼ Q 2 since it would imply M 2 L,R − M 2 a,b ∼ Q 2 . Indeed using previous values from [37] forM and M we obtain the estimate r ≃ 0.16, which may sufficiently support such an assignment parametrically.
Nevertheless, we argue that it is advantageous to keep the size of the ratio r open at first. This entails us to set up the expansion in a manner that permits to integrate out the 1 − fields efficiently. In order to connect to the chiral domain with m π ≪M − M , we must assume r ∼ Q 0 at least. Consistency of our results in that chiral domain will demand a further set of subtraction terms, as to eliminate power-counting violating contributions.
We turn to the bubble-type diagrams in Fig. 4 with where both types show a pole at s = M 2 H . In the derivation of the bubble-loop functions of Fig. 4 we need to separate their pole contribution first. The background and pole residuum terms can then be expanded according to the power-counting rules. We write and find the somewhat surprising expressions where we assumed r ∼ Q for simplicity. We return to such an assumption below in the context of the chiral expansion of triangle and box contributions. While dimensional counting rules suggest a leading contribution to B H ∼ Q 3 the specifics of such diagrams lead to terms of order Q 6 and higher only. Since we include terms up to order Q 4 only in this work, all such contributions can be dropped here. We note that the corresponding contributions to The bubble-type diagrams in Fig. 5 are with where we keep in (22) heavy tadpole terms proportional to r m 2 QĪ L /M 2 L and r m 2 QĪ R /M 2 R . Their scale dependence cannot be discriminated from the corresponding terms proportional to rĪ Q . In our scheme neither the LEC c 0 and c 1 nor g 1 receive a finite renormalization from the bubble loop terms in (22).

IV. SCATTERING WITH TRIANGLE DIAGRAMS
We turn to the triangle diagrams of Fig. 6 with where we use the conventional Mandelstam variables s, t and u of two-body scattering. The indices b and a specify the initial and final flavor channels of the chosen process. The three contributions in (23) correspond to the three rows in Fig. 6 in consecutive order. The first term is characterized by its s-channel, the second by its u-channel and the third by its tchannel unitarity cuts. For given isospin (I) and strangeness (S) channel the expressions can conveniently be factorized into universal loop functions and Clebsch coefficients, QH,R (s, u) , H,P Q (s, u) where the Clebsch coefficients depend on the isospin and strangeness of the intermediate The s-channel Clebsch coefficients are readily expressed in terms of the tree-level coeffi- where we map the channel index onto its meson content with c ↔ QH according to Tab. I.
The corresponding u-channel Clebsch C where we used identical summation indices with R → L in the first line of (24). One would expect that it is justified to neglect the mass differences from M L or M R in the loop functions summed over L or R. This leads to a factorization with the averaged Clebsch structures being implied by C u−ch as recalled already in (2). In particular we find with the Clebsch on the r.h.s of (27) already used in the Appendix of [37].
Similarly, the treatment of the t-channel terms is streamlined upon the introduction of symmetric and antisymmetric Clebsch and loop combinations in terms of the Clebsch listed in Tab. III and Tab. XXI of [37].
In the following we discuss in depth the computation of the loop functions. It suffices to specify the s-and t-channel loop functions. The u-channel expressions follow from the s-channel loop by the crossing replacement s ↔ u as is implied by q µ ↔ −q µ . We find where The renormalization scale of dimensional regularization is µ. Given the shortage of available letters in any notation scheme we purposely use µ in two distinct mathematical contexts.
From the specific form of the t-channel loop functions in (30) it is evident that they are all invariant under a simultaneous interchange of P ↔ Q and (q, p) ↔ (q,p), as was used in (28).
The proper evaluation of the triangle-loop functions in (30) is not quite so straightforward, in particular if one insists on the use of on-shell meson masses. Following previous works [51][52][53][54], in an initial step, our one-loop triangle contributions in (30) are expressed in terms of three scalar loop functions  (31). In case of the index Q the mass parameter m Q is encountered.
In the reduction of the triangle loops the following scalar bubble and triangle-loop ex-pressions occur in addition where we specify the kinematical points at which such integrals are needed.
There are a few well-known technical issues to be considered. A straightforward evaluation of the set of diagrams leads to results that suffer from terms that are at odds with their expected chiral power. There are terms, not only of too low, but also of too high orders, both of which need to be eliminated as to arrive at consistent results. For instance, according to dimensional counting rules one expects for properly renormalized scalar loop functions where we use a bar for renormalized quantities. As was already pointed out in [56] loop functions that are ultraviolet convergent do not give rise to power-counting violating contributions. Indeed, the expected chiral power of the scalar triangle loop can be confirmed by an explicit computation.
Yet, there is another technical complication that needs to be resolved. Any application of the original Passarino-Veltman decomposition scheme [55] requires the knowledge of specific correlations of the scalar basis functions at particular kinematic conditions [57][58][59][60][61][62][63][64]. If such relations are ignored results will suffer from kinematical singularities, a potentially pernicious situation. Therefore it is useful to extend the set of scalar basis integrals, such that a decomposition arises void of superficial singularities. This was advocated already in [53,54] in where we assure that both functions I (1) L,QH and I (1) QH,R are regular at the problematic threshold conditions s = (w · p) 2 /p 2 and s = (p · w) 2 /p 2 . The verification of our claim is tedious and asks for a more powerful viewpoint. We will generalize that extra basis functions with I (n) L,QH and I (n) QH,R , where with the case n = 0 we recover the original scalar triangles. We introduce the set of basis functions in terms of a Feynman parameter ansatz. We complement our choice of basis functions with or n > 1 expressions analogous to (34) can be derived, however, they turn more and more tedious as n increases, involving higher degrees of superficial pole structures. The explicit expression for n = 2 is given in (63) of Appendix A.
We note that while the integral representations (35) and (36)  We now assume that a given triangle loop is decomposed into our extended set of scalar loop functions. Such expressions are prohibitively involved, and therefore not shown here.
A useful reprentation can be obtained nevertheless upon a chiral expansion thereof. This goes in two steps. First we need to expand the coefficients in front of our basis functions in chiral powers. Here we apply the power-counting scheme [37] introduced in terms of on-shell hadron masses, as recalled in (15).
In order to specify the chiral order of a given contribution we need to assign a chiral power to the basis loop functions also. A subtraction scheme for the basis functions such that power-counting respecting renormalized basis loop functions arise is constructed. Such a procedure is symmetry conserving [53,56,65,66] as long as there is an unambiguous prescription how to represent such one-loop contributions in terms of the set of basis functions.
In this case we do not expect any violation of the chiral Ward identities of QCD.
Following our previous works [51][52][53][54] we introduce renormalized scalar bubble-loop functions that are independent of the renormalization scale. Here it is instrumental to carefully discriminate the light from the heavy particles.
where we use P, Q as placeholders for the light fields (Goldstone bosons) but H, L, R as placeholders for the heavy fields (charmed mesons). An explicit expression forĪ QH =Ī QH (s) was already recalled in (14). In turn it is left to renormalize the tadpole contributions with where we use s = M 2 and t = 0 together with H ∈ [1 − ] for simplicity.
We emphasize that here the introduction of the extended basis functions in (35)  After some algebra we arrive at the amazingly compact Q 3 terms in the triangle diagrams of Fig. 6. For the s-and u-channel diagrams it holds where we make the kinematical dependencies explicit again. The chiral counting rules (15) together with r ∼ Q are used. Given our renormalization scheme no power-counting vio- L,QH (s, u) ∼ Q 4 . This is a consequence of the scaling behavior of our basis functions (40) in that domain around s ∼ M 2 ab /2. We continue with the expansion of our t-channel loop functions in chiral powers according to (15) with r ∼ Q, where we drop terms only that are of order Q 4 or higher. With this we find the compact expressions where, again, the order Q 4 terms are shown in the Appendix. In (42)   Q,LR (s, u) loop, for which its coefficient in front of theĪ L andĪ R depends critically on terms of formally higher order. Via power-counting violating effects Any other choice would be at odds with this requirement.
The following Clebsch identities are useful in deriving the renormalization scale invariance of the sum of all third order terms. (1) with the Clebsch C (1) Q and C W T specifying the g 1 and g 2 terms in the third order tree-level contributions (1). The form of C We note that in the chiral domain we expect further suppressed results withJ

V. SCATTERING WITH BOX DIAGRAMS
We consider now the box diagrams of Fig. 7. The four contributions in (45) correspond to the two rows in Fig. 7 in consecutive order. The first term is characterized by its s-and tchannel, the second by its u-and t-channel unitarity cuts. The expressions can conveniently be factorized into universal loop functions and Clebsch coefficients, where the Clebsch coefficients depend on the isospin and strangeness of the intermediate We discuss the computation of the loop functions. It suffices to specify the two s-channel loop functions. The u-channel expressions follow from the s-channel loop by the crossing replacement s ↔ u as is already implied in (45). We find Our list of scalar integrals (31) and (35,36) needs an obvious extension with a scalar box integral Like in the case of the diagrams of Fig. 6 the proper evaluation of diagrams in Fig. 7 asks for an extension of the Passarino-Veltman functions. In Appendix C it is proven that the following set implies an unambiguous decomposition of the loop diagrams of Fig. 7   .
The two box-loop functions, as properly expressed in the particular set of basis functions, L,QH (s) QH,LR (s, t) (1)
Corresponding expressions at chiral order four can be found in Appendix D. The merit of our results rests on their compatibility with the expectation of power-counting rules, while keeping the on-shell meson masses throughout. Since the scalar basis functions are not further expanded our approximated renormalized expressions enjoy the correct analytic structure as it is requested in local quantum field theories from the micro-causality condition.
Since we started with un-renormalized expressions that suffer from large power-counting violating contributions it is absolutely crucial to eliminate the latter in a manner that is sufficiently effective so that a chiral expansion has convincing convergence properties.
While some readers may be worried about the complexity of our expressions, in particular the fourth order results in Appendix D, we note that a direct decomposition of (46) leads to more than a thousand terms, that cannot be properly expanded into chiral moments. Only with our novel scheme such contributions are cast into useful input for coupled-channel computations, the main target of our developments.

VI. SUMMARY AND OUTLOOK
In this work we studied triangle-and box-type contributions to two-body scattering in the context of the chiral Lagrangian with a heavy field. The formal developments are In the current work we overcome the above-described challenge by using an extended basis set, constructed such that kinematical constraints are avoided altogether and at the same time consistency with power-counting expectations is observed. We provided a proof that our decomposition is unique and exemplified our novel scheme with explicit expressions at chiral order three and four in the open-charm meson sector of chiral QCD.
In the next step we will use our results for an improved description of s-and p-wave scattering of Goldstone bosons off charmed meson states. This will be important for ongoing Lattice QCD computations on CLS ensembles, where owing to their large variety of β values a better control of discretization effects is expected. Here a quantitative success in the p-wave phase shifts may require the consideration of the left-hand cut contributions in the generalized potential as predicted by the chiral Lagrangian in terms of triangle and box contributions. Moreover, with our developments the path for an improved generalized potential approach to meson-baryon scattering based on the chiral Lagrangian is paved. In particular, the left-hand cut contributions can be extracted systematically from expressions as implied by our novel method.

APPENDIX A: SCALAR TRIANGLE LOOPS
We begin with an over-complete basis of scalar triangle-loop terms of the generic form into which each of the introduced diagram expressions (30) and (70) can be decomposed upon performing the contraction of the Lorentz indices. Without loss of generality we may assume f = k = i = 0 in the following. All other cases can be related to the particular choice study, where we assume Q µ =q µ + q µ . Such a reduction generates additional bubble-and tadpole-type integrals only, which do not cause any complications related to the introduction of our basis integrals (35) and (36).
The target function is analyzed in terms of a conventional Feynman parameter ansatz with F (x, y) = F G,LR (x, y) of (36) and some suitable real-valued coefficients The summation over the integers b, m, n start at zero. We split the integral into a convergent and scale-dependent piece with i J where we expand around d = 4. A further step shows that all scalar-triangle-type contributions take the form I 0 (m, n) with m + n ≤ a always. This is so since in the vicinity of d ∼ 4 it holds The remaining terms can be expressed in terms of bubble-type contributions. We assume the scale independent contributions as implied byÑ (x, y) to comply with the expectation of dimensional counting rules, while possible power-counting violating terms stem from the bubble-type contributions. They take the form where we celebrate the recursion relation 2 I (1) (1) that demonstrates our claim on the nature of such contributions.
The corresponding log terms involving F (x, 1 − x) in (56) follow upon the substitutions p →p − p and m G → M R in (58) and (59). In particular we find .
It remains to investigate the functions I 0 (m, n), for which we claim in (35) that it suffices to include a particular subset in our set of extended basis functions. This will be shown in the following by means of recursion equations that relate I 0 (m, n) for different choices of m and n.
We derive by suitable partial-integrations I 0 (m + 1, n) = −p · p p 2 I 0 (m, n + 1) +p QR − m dp 2 I 1 (m − 1, n) , I 0 (m, n + 1) = −p · p p 2 I 0 (m + 1, n) + which imply the desired recursions upon the elimination of the structure I 1 (m, n). The system (61) can be solved by iteration most economically. It is useful to consider first the case m = 0 for arbitrary n in the expressions I 0 (m + 1, n) for which the I 1 (−1, n) contribution vanishes identically. Given I 0 (0, n) we obtain all I 0 (1, n). Similarly, from the second equation in (61) we find I 0 (m, 1) from the set of all I 0 (m, 0) unambiguously. In the next step, we consider the second equation at m → m − 1 and n → n + 1, so that we can eliminate the common I 1 (m − 1, n) term from both equations. The resulting equation can be used to determine I 0 (m, n + 1) from I 0 (m, n) or alternatively I 0 (m + 1, n) from I 0 (m, n) by iteration.
Our basis functions in (35) are introduced with the particular choice I (n) G,LR = I 0 (n, 0) and I (n) G,LR = I 0 (0, n) in (52). Within such a scheme we derive for a (l · Q) in the numerator of (52) the following result where we observe that our result is invariant under the simultaneous exchange of L ↔ R andp ↔ p. It is emphasized that if and only if our result is expressed in terms of I 0 (0, 0) and I 0 (1, 0) (or I 0 (0, 1)) alone, a power-counting respecting expression is obtained with The proper evaluation of J G,LR with (l · Q) 2 in the numerator of (52) is slightly more tedious. It involves the additional basis functions I (2) G,LR and I (2) G,LR of (36) for which we derive an explicit representation LG − I (1) LG − I (0) which is an extension of (34). Our result (63) illustrates the necessity to include I (2) G,LR into our set of basis functions, as it is instrumental to avoid the kinematical singularity at (p · p) 2 /p 2 =p 2 . Note that from (36) it follows that I G,LR is regular at such kinematical conditions.
A direct application of (56) leads to a form for J (2) 000 that appears power-counting violating. The source of this complication is traced to its I 1 (0, 0) ∼ Q 2 contribution which should not be derived from (61). Instead, it is well-expressed in application of i J (0) 010 = I 1 (0, 0) + (p ·p) I 0 (2, 0) + 2 (p · p) I 0 (1, 1) + (p · p) I 0 (0, 2) where we observe that the particular combination does not involve the term I 1 (0, 0) by construction. As a consequence we find i J an expression that appears at odds with dimensional counting. From (66) we would see J (2) 000 ∼ Q 2 rather than the expected ∼ Q 4 . Our final expression ∼ Q 4 follows in application of (61), which leads to our result in terms of I LG − I which, in its renormalized form with in particularĪ p → w. Similarly, the QH, R case is implied by L → H withp → w. We note that our result can be readily generalized for the case defined by (l · Q) 2 → (l ·q) (l · q). It suffices to use the replacement Q µ Q ν → (q µ q ν + q µqν )/2 in (67).
Finally, it is advantageous in some cases, to use a symmetrized version of (67) that follows in application of the replacements L ↔ R andp ↔ p, under which J L,QH (s, u) QH,R (s, u) QH,R (s, u) QH,R (s, u) , where the missing u-channel term T for which we derive: We note that the loop functions J L,QH (s, u) in (71) upon the replacementsq ↔ q andp ↔ p and L ↔ R. An example for such a replacement is given in (41). It remains to detail the fourth order terms supplementing our third order expressions in (42). The following form is established where we use the convenient notation with m i ≥ 0 and o i ≥ 0 and n i ≥ 0 and h i ≥ 0 , x c , Again we split the integral into a convergent and scale-dependent piece with (q ·p) m 1 (q · w) o 1 (q · p) n 1 (p · q) m 2 (q · w) o 2 (p · q) n 2 × (p · p) h 1 (w 2 ) h 2 (q · q) h 3 C ab (x, z, y) −N ab (x, z, y) ∂ z F (x, z, y) , ∂ zLab (x, z, y) = L ab (x, z, y) , ∂ zNab (x, z, y) =Ñ (1) ab (x, z, y) , shows that all scalar-box-type contributions take the form I 0 (m, o, n) with m + o + n ≤ a + b always. This is so since in the vicinity of d ∼ 4 it holds where we used the third identity in (82). While the first line in (84) appears to cause a conflict with the expectation of power-counting, a suitable rewrite as implied by (82) leads to expressions that are consistent with this expectation after renormalization. Here we assume that the renormalized triangle loop functionsĪ H,LR → 0 vanish.
We turn to a (l · Q) 2 numerator in (74). Following our strategy already used successfully for the corresponding triangle case in (67), we manipulate that expression in two steps, with the first step involving the intermediate object B generalized for the case defined by (l · Q) 2 → (l ·q) (l · q). Like in the corresponding triangle case it suffices to use the replacement Q µ Q ν → (q µ q ν + q µqν )/2 in (86).