Symmetry preserving truncations of the gap and Bethe-Salpeter equations

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by  Daniele Binosi

Symmetry preserving truncations of the gap and Bethe-Salpeter equations
Daniele Binosi,1 Lei Chang,2 Joannis Papavassiliou,3 Si-Xue Qin,4 and Craig D. Roberts4
1European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT∗) and Fondazione Bruno Kessler
Villa Tambosi, Strada delle Tabarelle 286, I-38123 Villazzano (TN) Italy
2School of Physics, Nankai University, Tianjin 300071, China
3Department of Theoretical Physics and IFIC, University of Valencia and CSIC, E-46100, Valencia, Spain
4Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
(Dated: 19 January 2016)
Ward-Green-Takahashi (WGT) identities play a crucial role in hadron physics, e.g. imposing
stringent relationships between the kernels of the one- and two-body problems, which must be
preserved in any veracious treatment of mesons as bound-states. In this connection, one may view
6 thedressed gluon-quark vertex,Γaµ, as fundamental. We usea novel representation of Γaµ, in terms
1 ofthegluon-quarkscatteringmatrix,todevelopamethodcapableofelucidatingtheuniquequark-
0 antiquark Bethe-Salpeter kernel, K, that is symmetry-consistent with a given quark gap equation.
2 A strength of the scheme is its ability to expose and capitalise on graphic symmetries within the
kernels. Thisisdisplayedinananalysis thatrevealstheorigin of H-diagramsin K,which aretwo-
n
particle-irreduciblecontributions, generated astwo-loop diagrams involvingthethree-gluon vertex,
a
thatcannotbeabsorbedasadressingofΓa inaBethe-Salpeterkernelnorexpressedasamemberof
J µ
theclassof crossed-boxdiagrams. Thus,therearenogeneral circumstancesunderwhichtheWGT
0
identities essential for a valid description of mesons can be preserved by a Bethe-Salpeter kernel
2
obtained simply by dressing both gluon-quark vertices in a ladder-like truncation; and, moreover,

addingany numberof similarly-dressed crossed-box diagrams cannot improve thesituation.
]
h
t PACSnumbers: 11.10.St;11.30.Rd;12.38.Aw;12.38.Lg
-
l
c
u 1. Introduction. A natural framework for studying ter alia, that all Ward-Green-Takahashi (WGT) identi-
n
the two valence-body bound-state problem in quantum ties [4–7] are preserved, without fine-tuning, and hence
[
field theory is provided by the Dyson-Schwinger equa- ensures, e.g. current-conservationand the appearance of

1 tions (DSEs) [1], with the one-body gap equation and Goldstones modes in connection with DCSB.
v two-body Bethe-Salpeter equation (BSE) playing lead- The leading-orderterm in the procedure of Refs.[2, 3]
1 ing roles. The approach is useful in hadron physics ow- is the rainbow-ladder (RL) truncation. It is widely
4
ing to asymptotic freedom in quantum chromodynam- used and known to be accurate for light-quark ground-
4
ics (QCD), which materially reduces model dependence state vector- and isospin-nonzero-pseudoscalar-mesons
5
0 in sound nonperturbative applications because the in- [8–11],andpropertiesofground-stateoctetanddecuplet
. teraction kernel in each DSE is known for all momenta baryons [12–15], because corrections in these channels
1
within the perturbative domain, i.e. k2 & 2GeV2. Any largelycancelowingtotheparameter-freepreservationof
0
model need then only describe the kernels’nonperturba- relevant WGT identities ensured by this scheme. How-
6
1 tive behaviour. That is valuable because DSE solutions ever, higher-order contributions do not typically cancel
: are propagatorsandvertices,in terms of whichallcross- in other channels [16–18]. Hence studies based on the
v
sections are built. The approach thus connects observ- RL truncation, or low-order improvements thereof, usu-
i
X ables with the long-range behaviour of QCD’s running allyprovidepoorresultsforlight-quarkscalar-andaxial-
r coupling and masses. Hence, feedback between predic- vector-mesons [19–24], exhibit gross sensitivity to model
a
tions and experimental tests can be used to refine any parametersfortensor-mesons[25]andexcitedstates[26–
modelinputandtherebyimproveunderstandingofthese 29], and are unrealistic for heavy-light systems [30–32].
basic quantities. This opens the way to addressingques- These difficulties are surmounted by the scheme in
tions pertaining to, e.g.: the gluon- and quark-structure Ref.[33] because it enables the use of more realis-
ofhadrons;andtheemergenceandimpactofconfinement tic kernels for the gap and Bethe-Salpeter equations,
and dynamical chiral symmetry breaking (DCSB). which possess a sophisticated structure, including Dirac
The DSEs are a collection of coupled equations; and vector⊗vector and scalar⊗scalar quark-antiquark inter-
a tractable problem is only obtained once a truncation actions. Significantly, this technique is also symmetry
scheme is specified. A weak-coupling expansion repro- preserving; but it does not require knowledge of the di-
duces perturbation theory; but, although valuable in agrammatic content of the gap equation’s kernel, whose
the analysis of large momentum transfer phenomena in complexity may be expressed in the form chosen for the
QCD, it cannot yield nonperturbative information. A dressed gluon-quark vertex, a subject of great interest
symmetry-preserving scheme applicable to hadrons was itself, e.g. Refs.[34–42]. The gap equation in Fig.1 is:
introduced in Refs.[2, 3]. That procedure generates a
BSE from the kernel of any gap equation whose dia- Σ(k)=Z Λg2D (k−q)Γa(k,q)S(q)λaγ , (1)
grammatic content is known. It thereby guarantees, in- 1Z µν µ 2 ν
dq
2
FIG. 1. Quark self-energy, Σ(k)=iγ·k[A(k2)−1]+B(k2),
Eq.(1): solid line with open circle, dressed-quarkpropagator
S(q) =1/[iγ·q+Σ(q)]; open-circle “spring” , dressed-gluon
propagator,Dµν(p−q);and(red)shadedcircle,dressedgluon-
quark vertex,Γa(q,p). A coupling g appears at each vertex.
µ
Λ
where represents a Poincar´e invariant regularisation
dq FIG. 2. Upper panel. BSE for a colour-singlet vertex,
of the Rfour-dimensional integral, with Λ the regulariza-
tion mass-scale, and Z1(ζ2,Λ2), is the vertex renormal- wGMith(kg;MP)=. T21τhieγ5chγµanonneelgisaidnesfiancecdesbsytothaellisntahtoemsothgeanteciotym,me.ug-.
isation constant, with ζ the renormalisation scale. An
nicate with an isovector axial-vector probe, such as the pion
additional strength of the new scheme is its capacity to and a1 meson. The interaction between the dressed valence-
express DCSB in the integral equations connected with constituentsiscompletely described bythescattering kernel,
bound-states. It has therefore enabled elucidation of K. Lowerpanel. ForI 6=0mesons,theBethe-Salpeterkernel
novel nonperturbative features of QCD [13, 38, 43–46] isasumoftwoterms,Eq.(4). Theinteractioncontentofboth
andfacilitatedacrucialsteptowardthe ab initio predic- is completely determined by that of the dressed gluon-quark
tion of hadron observables in continuum-QCD [47]. vertex,explicitly for KS and implicitly for KΓ, Eq.(5).
Notwithstanding the existence of this improved
scheme, there is merit in providing a mechanical ap-
antiquark line. Naturally, this means that K also in-
proachcapable of elucidating that Bethe-Salpeter kernel
cludes an enumerable infinity of n≥2-PI contributions.
which is symmetry-consistent with any given gap equa-
The kernelthatensurespreservationofallWGT iden-
tion. Possessing such a tool, one may, e.g. validate any
tities associated with a given colour-singlet vertex may
newly-proposedBethe-Salpeter kernelthat is based on a
be expressed as [2, 3]:
skeletonexpansionofthe gapequation,simplyby check-
ing whether it ensures preservation of the WGT identi- δΣ(k)
K(k,q;P)=− , (3)
ties, and/or expose the full complexity demanded of the δS(q)
symmetry-consistent kernel by any concrete statement
aboutthe gapequation’s structure. We describe suchan whichcorrespondsto“cutting”eachinternalfermionline
approach herein, delivering our explanations mainly in inalldressingdiagrams. Thiscuttingprocedureactually
terms of diagrams. Naturally, each one corresponds to furnishes a kernel in the “diagonal configuration” [48]:
a well-definedintegral,whichcouldbe writtenexplicitly. K(k,q;0). The general momentum configuration may
In those terms, however, the proliferation of symbols, be obtained as described following Eq.(44) in Ref.[17],
nested integrals, etc. would obscure the reasoning. In i.e. in addition to the usual effect of differentiation, the
using diagrammatic methods we are capitalising on the functional derivative adds P to the argument of every
pedagogical capacity and intuitive strengths which have quark line through which it is commuted when applying
led to Feynman diagrams being adopted so widely. the product rule. One may alternatively generate the
full momentum arguments by beginning with the effec-
2. Insufficiency of vertex-dressed ladder kernels.
tive action A[S] expressed in coordinate space [49, 50],
The BSE for a colour-singletvertex, G , which may ex-
M obtaining the dressed-quark propagator as the solution
hibit meson bound-states, is depicted in Fig.2:
of δA[S]/δS =0, and subsequently employing Eq.(3).
[G (k;P)] =Z g + For systems with nonzero isospin, I 6=0, quark propa-
M rs M M
gatorsappearing in gluon vacuum-polarisationdiagrams
Λ
[S(q )G (q;P)S(q )] Krs(k,q;P), (2) may be neglected and the kernel can be expressed as a
Zdq + M − tu tu sum of just two terms [3, 17, 18, 33, 52, 53]:
where Z is a renormalisation constant, the total mo- KI6=0(k,q;P)=K (k,q;P)+K (k,q;P), (4)
M S Γ
mentum P = k − k , where k = k + ηP, k =
+ − + −
as depicted in the lower panel of Fig.2, where the con-
k−(1−η)P, with η ∈ [0,1]: no observable can depend
tent of the vertex Λ is completely determined by the
on η, i.e. the definition of the relative momentum. µ
The scattering kernel, K(k,q;P) in Eq.(2), expresses functional derivative of the dressed gluon-quark vertex:
allpossibleinteractionsthatcanoccurbetweenadressed δΓa(k,q)
quark and dressed antiquark; and is two-particle irre- Λaµ(k,q;P)∼ δµS(q) . (5)
ducible(2PI),viz.itdoesnotcontainquark+antiquarkto
single gauge-boson annihilation diagrams nor diagrams Extending our reasoning to I = 0 systems is not diffi-
that become disconnected by cutting one quark and one cult in principle; but many extra diagramsarise and one
3
0.8
0.6 L-γΓ
L-ΓΓ
Full
") 0.4
(
’
& 0.2
"$$%!# 0
−0.2
−0.4
0 20 40 60 80 100
*+$%!’()
FIG. 3. Pion mass-squared vs. current-quarkmass, obtained
with the BC Ansatz, Eq.(6), for Γa in the gap equation,
µ
Fig.1: dashed (blue) curve, Bethe-Salpeter kernel in Fig.2
is K = KS; dot-dashed (red) curve, kernel is KL-ΓΓ, the
dressed-vertex ladder-like kernel; and solid (black) curve,
complete BC-vertex-consistent kernel constructed following
Refs.[33,51]. Evidently,onlythecompletekernelissufficient
toensuretheexistenceof aNambu-Goldstonepion. (Results
obtainedwiththedressed-gluonlinerepresentedbytheinter- FIG.4. Upperpanel. DSEforthedressedgluon-quarkvertex,
action in Ref.[28]: D=0.5GeV2,ω=0.5GeV, τ →∞.) Γa. Only selected contributions are shown. The complete
µ
equation is depicted, e.g. in Fig.2.6 of Ref.[1]. The shaded
(blue)circleatthejunctionofthreegluonlinesisthedressed
must also allow for the possibility that contributions of three-gluon vertex. Lower panel. Some of the contributions
a topological nature may play an important role [54]. tothequark-antiquarkscatteringkernel,K,generatedbythe
In some exceptional circumstances, as when Γa is ex- gap equation expressed in terms of the dressed gluon-quark
µ
vertex depicted in the upper panel. The last diagram drawn
pressed via a recursion relation (an integral equation
explicitlyisanexampleofanH-diagram. Itis2PI;butcannot
whose kernelhas a simple, closedform) andthe dressed-
beexpressedasacorrectiontoeithervertexinaladderkernel
gluon propagator has negligible support at nonzero mo-
noras a memberof theclass of crossed-box diagrams.
menta, e.g. the model in Ref.[55], Λa ≡ 0 in the pseu-
µ
doscalarchannel[17,18]. Inthis casetheBethe-Salpeter
kernel that preserves the axial-vector WGT identity for
diagraminFig.4andits analoguesinotherrelevantker-
a given gap equation with dressing defined by Γa, is ob-
µ nels. (Abelian-like contributions may be subdominant
tained by including the specified dressing on only one of
[18, 35, 57].) One then arrives at the diagrams in the
theverticesinaladder-likekernel,viz.K (k,q;P). That
S secondline ofFig.4. Insertingthis vertexintoFig.1and
isnottrueinanyotherchanneland,ingeneral,falseinall
using Eq.(3), it is apparent that the first contribution
channels; a fact emphasised by the dashed (blue) curve
(bare gluon-quark vertex) generates the RL truncation,
in Fig.3, which displays the pion mass obtained using a
which is the first term depicted on the right-hand-side
Ball-Chiu (BC) vertex Ansatz [56] (t=[k+q]/2):
(rhs) of the lower panel in Fig.4. It is evident, too, that
λa the second diagram in the second line of Fig.4 produces
iΓaµ(k,q)= 2 ςAiγµ+δA2itµγ·t+δBtµID , (6) a sum of two terms in K, viz. a three-gluon vertex cor-
(cid:2) (cid:3) rection on the quark line and another on the antiquark
ς = [F(k2) + F(q2)]/2, δ = [F(k2) − F(q2)]/(k2 − line. These are the second two terms in the lower panel
F F
q2), for the dressed vertex in Fig.1 and K = K in the of Fig.4; therefore, at this point it might seem plau-
S
upperpanelofFig.2. Plainly,althoughthegapequation sible that a vertex-dressed ladder kernel can provide a
guaranteesDCSB,theBSEdoesnotproduceapionwith symmetry-preservingframeworkforthestudyofmesons.
the characteristics of a Goldstone-boson. However, the third diagram drawn in the second line
One might imagine that if dressing only one vertex of Fig.4 has not yet been considered. Amongst others,
fails, then, perhaps, dressing both vertices, to obtain a it generates an H-diagram, viz. the last image drawn
dressed-vertex ladder-like kernel, KL-ΓΓ, will be suffi- in the lower panel of Fig.4. Kindred contributions to
cient to produce a symmetry-consistent system. It is Γa produce an enumerable infinity of similar terms in
µ
known from Refs.[3, 17, 18, 33] that this is false in gen- K. Such contributions cannot be expressed as a correc-
eral: the simplest Abelian-like one-loop gluon-correction tion to either vertex in a ladder kernel and neither are
to the gluon-quark vertex (generated by the diagram la- they members of the class of crossed-box diagrams. H-
belled “Ab” in Fig.4) demands the presence of crossed- diagrams are a distinct class of essentially non-Abelian
box contributions to the symmetry-consistent kernel. 2PI contributions to K. If they are omitted from the
Suppose though, that one neglects Abelian-like dress- kernel, then the BSE obtained thereby cannot produce
ings of Γa; namely, all terms generated by the Ab- colour-singletverticesthatsatisfyWGTidentitiesinvolv-
µ
4
ing a dressed-quark propagator generated by the vertex
inFig.4,whetherornotAb-typediagramsareneglected.
ItisnotablethatH-typediagramsproduceaninfrared
divergence in the perturbative calculation of the static-
quark potential, i.e. a contribution which exhibits un-
bounded growth as the distance between the source and
sink increases [58]. One might view this as a sign that
the inclusion of H-type diagramsin the quark-antiquark
scattering kernel could be important if one seeks to re-
FIG.5. Upperpanel. DSEforthedressedgluon-quarkvertex,
cover an area-law in the static-quark limit [59].
Γa,expressed withtheantiquarkprovidingthereference line
The general insufficiency of the vertex-dressed ladder µ
and involving the s-channel 1PI gluon-quark scattering am-
kernel(KL-ΓΓ)isalsohighlightedbyFig.3: withtheBC plitude C. Lower panel. In terms of the gluon-quark vertex,
vertex, Eq.(6), one obtains the dot-dashed (red) curve. theequationforthequarkself-energyismanifestlysymmetric
Plainly, this kernel generates far too much attraction. when expressed using C.
That is not surprising, given the dashed (blue) curve in
the same figure, which shows that K provides attrac-
S
tioninthepseudoscalarkernel;and,indressingbothver- gluon-quark vertex. Namely, instead of considering the
tices,onehasnotobviouslyaddedanyrepulsion. Impor- vertex from the gluon’s perspective, it is advantageous
tantly,however,there isactuallydestructiveinterference to adopt the antiquark’s view, depicted in Fig.5, and
amongst the various contributions in the BSE obtained write a DSE for this vertex in terms of the gluon-quark
with KL-ΓΓ: whereas the ςA2 terms produce attraction, scattering amplitude C, which is 1PI in the s-channel:
those involving δB generate net repulsion. Γa(p,q)= λaγ +ΓQ =:Γ(0)∗M , (8)
Considertherefore,amodifiedAnsatz,viz.Eq.(6)with µ 2 µ µ µ
Λ
δB → 2δB. In this case, weakening the interaction ΓQ = λbγ S(ℓ )D (ℓ )Cba(ℓ,q;p), (9)
strength: D = 0.5 → 0.29GeV2 produces m2 = 0 at µ Z 2 ρ + ρσ − σµ
π dℓ
m =0. Evidently,onecantunetheinteractionstrength
toqachieve a massless pion; but securing that numerical where Γ(µ0) = λ2aγµ is the tree-level contribution, ΓQµ ex-
outcomeisnotequivalenttoensuringpreservationofthe presses all (quantum) corrections, and the sum is repre-
axial-vectorWGTidentity. Thisisreadilyseenbycheck- sented by the operation of the transition matrix M, de-
ing the quark-levelGoldberger-Treimanrelation[60,61]: fined implicitly in Eq.(8). Inserting Eq.(9) into Eq.(1),
one obtains the manifestly symmetric expression for the
E (k2,k·P =0;P2 =0)m∝q=0B(k2), (7) quark self-energy depicted in the lower panel of Fig.5.
π Eq.(3)cannowbeusedtoobtainthatkernelinEq.(2)
whichensurespreservationofallWGTidentitiesrelevant
a corollary of the axial-vector WGT identity, where E
π to the channel considered. For I 6=0, the differentiation
is the leading term in the pion’s Bethe-Salpeter ampli-
produces the series in Fig.6. Once more, the rainbow-
tude. Using the δ → 2δ Ansatz, both in the gap
B B ladder truncation appears as the simplest contribution,
equation and to generate KL-ΓΓ, one finds that Eq.(7) arisingfromdiagram(a)inFig.5;butitisaugmentedin
is violated: an accurate interpolation of the monoton-
general by a series of complex corrections. Denoting the
icall decreasing ratio is provided by E (k2)/B(k2) =
(1+0.02x)/(1+0.08x+0.01x2),x∈[0,50]π,x=k2/B2(0), thirddiagraminline(b)ofFig.6byΣ6C(k),viz.theimage
withC itselfbeingcut,andrepresentsitscontributionin
where B(k2 = 0) = 0.28GeV. (We normalised the ratio
to unity at k2 = 0. It is 0.47 at x = 9.) On the other the BSE by K6C, then it is evident from Fig.6 that
hand, Eq.(7)is preservedwithout fine tuning whenK is KI6=0 =−g2 Γ(0)D Γ(0)
µ µν ν
constructed according to Ref.[33]. It follows that there
(cid:2)
exist gap equation kernels, too numerous to count, for +Γ(µ0)DµνΓQν +ΓQµ DµνΓ(ν0) +K6C (10)
which KL-ΓΓ yields m2π = 0 at mq = 0; but the pairing =−g2ΓµDµνΓν+g2ΓQµDµνΓQν (cid:3)+K6C . (11)
nevertheless fails to preserve the axial-vector identity.
The preceding discussion invalidates claims made in KL−ΓΓ
| {z }
Ref.[62], e.g., referring now to diagrams therein, under Plainly, KL-ΓΓ in Eq.(11) is a dressed-vertex ladder-
no circumstances can the BSE in Fig.12 be symmetry-
likesubcomponentofthesymmetry-consistentkernel(al-
consistent with the gap equation generated by Fig.5.
ready considered in Sec.2); but there is much more in
3. Symmetry-consistent Bethe-Salpeter Kernel. the complete kernel. To elucidate, we focus on K6C and
InordertogeneralisethediscussioninSec.2,wefirstob- consider the nature of C. This amplitude contains in-
serve that the common manner of expressing the quark finitely manydiagrams;andalthoughaninfinite number
self-energy, Fig.1, is grossly asymmetric with respect to of them contain no internal quark line, that collection
the twogluon-quarkvertices: one vertexis fully-dressed, cannot contribute to K6C. The relevant contributions to
whereas the other has its tree-level form. This can be C are those which contain at least one quark line; and
remedied by changing the way one looks at the dressed hence,forI 6=0,K6C hastheexpansiondepictedinFig.7.
5
FIG.8. ExplicatingtheoriginofH-diagramsandcrossed-box
terms in thequark-antiquarkscattering kernel, KI6=1. Left –
elementsinthegluon-quarkscatteringmatrix,C;andright –
FIG. 6. In I 6= 0 channels, Eq.(3) produces this series of contributions theygenerate in K6C.
diagramsforthe2PIkernelofthesymmetry-consistentBSE.
Thedashed(red)linesrepresenttheactoffunctionaldifferen-
tiationandthearrowsdirectattentiontotheresultingkernel the top-left image expresses a correction to the gluon-
contribution, when that can bedepicted simply. quark vertex, so its influence is felt within KL-ΓΓ; but
itmustsimultaneouslycontributetoK ,asdisplayedin
6L
Fig.7, producing the H-diagramdepicted Fig.8.
4. Epilogue. Working with a simple Ansatz for the
dressedgluon-quarkvertex,Γa,weconsideredthecapac-
µ
ityofvertex-dressedladder-likeBethe-Salpeterkernelsto
preserve Ward-Green-Takahashi (WGT) identities rele-
vant to meson bound-states; and found that whilst they
can readily be tuned to produce a massless pion in the
chiral limit, they nevertheless fail to preserve the axial-
vector WGT identity and are thus incomplete. We gen-
eralised these observations using a novel representation
of Γa in terms of the gluon-quark scattering amplitude,
µ
which enabled us to show that whilst a dressed-vertex
ladder-like truncation is the simplest term in the com-
plete Bethe-Salpeter kernel, K, it is insufficient in gen-
FIG.7. Upperpanel. InI 6=0channels,K6C hastheexpansion eral,owingtothepresenceof,interalia,H-diagrams,viz.
depictedhere,wherethedashed(red)linesindicatethequark two-loop terms, involving the three-gluon vertex, that
line that reacts to the functional differentiation in Eq.(3). cannotbeabsorbedintoK asapartofΓa norexpressed
µ
The expansion necessarily involves g +q → (m+1)g +q, asamemberoftheclassofcrossed-boxdiagrams. Conse-
m ≥ 1, transition matrices, M1, etc. Lower panel. A focus quently, the WGT identities essentialfor a validdescrip-
onthefirstdiagramontherhsintheupperpanelrevealsthe tion of mesons cannot generally be preserved when K
simple nature of its contribution to K.
is obtainedmerelyby dressingbothgluon-quarkvertices
in a ladder-like truncation; and, moreover, adding any
numberofsimilarlydressedtermsintheclassofcrossed-
The lower panel of Fig.7 focuses on the first term
boxdiagramscannotimprovethesituation. Fortunately,
on the rhs of the upper panel, which produces the
sophisticated alternatives exist [33], are in practical em-
following contribution to the quark-antiquark kernel:
ployment [38, 43–47] and are being refined [51].
(−g2)ΓQD ΓQ. All remaining terms generate n ≥ 2-
µ µν ν
PI contributions to KI6=0 that are structurally inequiva- Acknowledgments. We are grateful for construc-
lenttothosecontainedinKL-ΓΓ,asillustratedbyFig.8. tive remarks from A. Bashir, S.J. Brodsky, I.C. Clo¨et,
Denoting these terms by K , one arrives finally at B. El-Bennich, G. Krein, C. Mezrag, J. Rodr´ıguez-
6L
Quintero,andJ.Segovia. Researchsupportedby: Span-
KI6=0 =KL-ΓΓ+K6L. (12) ish MEYC grant nos. FPA2014-53631-C2-1-Pand SEV-
2014-0398; Generalitat Valenciana under grant Prome-
That it is impossible for KL-ΓΓ alone to serve as a teoII/2014/066; Argonne National Laboratory Office of
symmetry-consistentquark-antiquarkscatteringkernelis the Director, through the Named Postdoctoral Fellow-
also evident here. Amongst infinitely many others, the ship Program; and U.S. Department of Energy, Office
second and third images drawn explicitly in the expres- of Science, Office of Nuclear Physics, contract no. DE-
sion for K6C in Fig.7 contain the contributions in Fig.8: AC02-06CH11357.
6
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