11institutetext: Institute for High Energy Physics, Austrian Academy of
Sciences, Nikolsdorfergasse 18,
A-1050 Vienna, Austria22institutetext: D. V. Skobeltsyn Institute of Nuclear
Physics, M. V. Lomonosov Moscow State University,
119991 Moscow, Russia33institutetext: Joint Institute for Nuclear Research,
141980 Dubna, Russia44institutetext: Faculty of Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna, Austria55institutetext: Université Paris-
Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

# Zooming in on Multiquark Hadrons
within QCD Sum-Rule Approaches

Wolfgang Lucha 11<EMAIL_ADDRESS>Dmitri Melikhov 223344
<EMAIL_ADDRESS>Hagop Sazdjian 55<EMAIL_ADDRESS>
###### Abstract

Aiming at self-consistent descriptions of multiquark hadrons (such as
tetraquarks, pentaquarks, hexaquarks) by means of QCD sum rules, we note that
the totality of contributions to two-point or three-point correlation
functions that involve, respectively, either two or just a single operator
capable of interpolating the particular multiquark under study can be
straightforwardly disentangled into two disjoint classes defined by
unambiguously identifiable members. The first is formed by so-called
multiquark-phile contributions which indeed might support multiquarks. In the
case of flavour-exotic tetraquarks, by definition composed of four (anti-)
quarks of mutually different flavours, a tetraquark-phile contribution has to
exhibit two or more gluon exchanges of appropriate topology. The second
consists of contributions evidently not bearing any relation to multiquarks;
these must be discarded when studying multiquarks by QCD sum rules. The first
class only should enter the “multiquark-adequate” QCD sum rules for exotic
hadrons.

##  1 Multiquark Hadrons, Subset of Unconventional QCD Bound States

The relativistic quantum field theory that provides the fundamental-level
description of strong interactions in terms of quarks and gluons acting as
fundamental degrees of freedom, quantum chromodynamics (QCD), in principle
permits, as bound states of these degrees of freedom, all combinations that
form a singlet under its non-Abelian gauge group SU(3). Therefore, among its
bound states there have been observed not only the two categories of
_conventional_ hadrons, quark–antiquark mesons and three-quark baryons, but
even (candidates for) representatives of various, likewise QCD-compatible
categories of so-called nonconventional or _exotic_ hadrons. To these belong
_multiquark_ hadrons (tetraquarks, pentaquarks, hexaquarks, heptaquarks, …),
quark–gluon bound states, called hybrid, and pure-gluon bound states,
(nick)named glueballs.

Within the framework of quantum field theory, the concept of QCD sum rules QSR
provides a nonperturbative analytic (that is, not merely numerical) approach
to bound states. QCD sum rules represent a means to _comparatively_ easily
trace back observable properties of hadrons to all the degrees of freedom and
parameters entering the underlying quantum field theory, QCD. Quite generally,
their derivation proceeds along a sequence of manipulations the main steps of
which may be outlined by the following recipe: Identify or, if necessary,
construct an operator that interpolates the hadron in your focus of interest,
by having a nonvanishing matrix element if sandwiched between the vacuum and
the particular hadron state, from the pool of quark and gluon field operators
offered by QCD. Consider an apparently convenient correlation function of
hadron interpolating operators. Evaluate the correlation function at both
phenomenological hadron and fundamental QCD level. To this end: Apply the
operator product expansion KGW , in order to separate perturbative from
nonperturbative contributions. Allow the hadron spectrum to enter the game by
inserting a complete set of hadron states. Emphasize lowest-mass hadron states
by utilizing fitting Borel transformations. Assume that, above (ideally
optimized ET1 ; ET2 ; ET3 ) effective thresholds, perturbative-QCD and hardly
known higher-hadron contributions cancel.

Our present intention is to lift the standard QCD sum-rule formalism encoded
in the above prescription – in particular, the theoretical starting point,
viz., the QCD input to the correlation function – to a level better matching
the group-theoretically induced peculiarities of the exotic multiquark hadrons
that strikingly distinguish them from the class of ordinary hadrons ESRp ;
ESRr ; TMA1 ; TMA2 ; LMS10 ; TMA3 . Specifically, for well-defined categories
of tetraquarks LMS4 , we are led to suspect that this goal necessitates a QCD
contribution to involve two or more gluon exchanges of suitable topology.

Below, we recall in brief some of our earlier considerations ESRp ; ESRr ;
LMS10  on how to adequately optimize the QCD sum-rule approach to multiquark
states. In this context, the main challenge is to identify and subsequently
single out precisely those contributions to correlation functions that have
the potential to influence QCD sum-rule predictions of multiquark features
(Sect. 2). Upon following the route materializing thereby, anyone’s quest for
increased precision should be rewarded by eventually arriving at so-called
multiquark-adequate QCD sum rules (Sect. 4), as illustrated, for clarity, for
the tetraquarks with “flavour-exotic” quark composition (Sect. 3). Ignoring
all for the subsequent line of argument irrelevant issues (like parity or spin
degrees of freedom, by notationally suppressing Dirac matrices) will enable us
to focus on the essentials.

##  2 Tetraquark Mesons, Conceptually Simplest Form of Exotic Hadron

At present, the – at least from the experimental perspective PDG  – presumably
best established category of exotic multiquark hadrons seems to be formed by
the totality of tetraquark mesons

$T=[\overline{q}_{a}\,q_{b}\,\overline{q}_{c}\,q_{d}]\ ,$ (1)

bound states of two quarks $q_{b},q_{d}$ and two antiquarks
$\overline{q}_{a},\overline{q}_{c}$ carrying flavour quantum numbers

$a,b,c,d\in\\{u,d,s,c,b\\}$ (2)

and having masses $m_{a},m_{b},m_{c},m_{d}$ that (like the strong coupling)
are _basic parameters_ of QCD and will assume a prominent rôle in the
tetraquark-adequate QCD sum rules drafted in Sect. 4.

Following, for the intended construction of the appropriate tetraquark-
adequate QCD sum rules, the prescription recapitulated above, we first have to
specify the tetraquark interpolating operators we would like to employ. Our
task is significantly facilitated by the observation RLJ  that, by application
of suitable Fierz transformations MF , any chosen tetraquark interpolating
operator may be demonstrated to be equivalent to a linear combination of
merely two products

$\theta_{\overline{a}b\overline{c}d}(x)\equiv
j_{\overline{a}b}(x)\,j_{\overline{c}d}(x)\
,\qquad\theta_{\overline{a}d\overline{c}b}(x)\equiv
j_{\overline{a}d}(x)\,j_{\overline{c}b}(x)$ (3)

of SU(3)-singlet quark–antiquark bilinear operators $j_{\overline{a}b}(x)$
interpolating conventional mesons,

$j_{\overline{a}b}(x)\equiv\overline{q}_{a}(x)\,q_{b}(x)\ .$ (4)

(Useful operator identities are provided by Ref. LMS111 , Eqs. (32), (36), or
Ref. ESRr , Eqs. (1), (2).)

Since having in mind to unveil the deepest secrets of the tetraquark mesons by
scrutinizing their effects upon contributing, by way of intermediate-state
poles, to amplitudes encoding the scattering of two conventional mesons into
two conventional mesons, a smart idea is to trust in correlation functions of
a time-ordered product of four quark–antiquark bilinear operators (4),

$\left\langle{\rm
T}\\!\left(j(y)\,j(y^{\prime})\,j^{\dagger}(x)\,j^{\dagger}(x^{\prime})\right)\right\rangle.$
(5)

Configuration-space contractions of both pairs of quark–antiquark bilinear
operators (4) yield correlation functions of two tetraquark interpolating
operators (3) (cf. Subsect. 3.1.1 or 3.2.1):

$\left\langle{\rm
T}\\!\left(\theta(y)\,\theta^{\dagger}(x)\right)\right\rangle=\lim_{\underset{\scriptstyle
y^{\prime}\to y}{\scriptstyle x^{\prime}\to x}}\left\langle{\rm
T}\\!\left(j(y)\,j(y^{\prime})\,j^{\dagger}(x)\,j^{\dagger}(x^{\prime})\right)\right\rangle.$
(6)

Configuration-space contractions of one pair of quark-bilinear operators (4)
create correlation functions involving only one tetraquark interpolating
operator (3) (cf. Subsect. 3.1.2 or 3.2.2):

$\left\langle{\rm
T}\\!\left(j(y)\,j(y^{\prime})\,\theta^{\dagger}(x)\right)\right\rangle=\lim_{x^{\prime}\to
x}\left\langle{\rm
T}\\!\left(j(y)\,j(y^{\prime})\,j^{\dagger}(x)\,j^{\dagger}(x^{\prime})\right)\right\rangle.$
(7)

Now, from our point of view, at least, the decisive move towards the envisaged
multiquark adequacy TMA2  of QCD sum rules is the unambiguous sorting out of
all QCD-level contributions that promise to bear potential relevance for
multiquarks from those that, beyond doubt, do not. Accordingly, in order to
sharpen our blades, we formulated, for the case of tetraquark mesons, a simple
and, in fact, rather self-evident criterion TQC1 ; TQC2  that enables us to
separate the wheat from the chaff by identifying those contributions,
henceforth named _tetraquark-phile_ TQP1 ; TQP2 , that may contribute to the
expected formation of some tetraquark pole: Let $p_{1},p_{2}$ and
$q_{1},q_{2}$ be the four external momenta related to a generic correlation
function (5). Recall the definition of the Mandelstam variable $s$ in terms of
initial and final momenta $p_{1},p_{2}$ and $q_{1},q_{2}$, respectively,

$s\equiv(p_{1}+p_{2})^{2}=(q_{1}+q_{2})^{2}\ .$ (8)

We deem a QCD-level contribution, considered as function of $s$, _tetraquark-
phile_ if it exhibits a nonpolynomial dependence on $s$ and if it develops a
branch cut that starts at the branch point

$\hat{s}\equiv(m_{a}+m_{b}+m_{c}+m_{d})^{2}\ .$ (9)

Whether or not the requirement of the existence of an appropriate branch point
is fulfilled may easily be decided by inspection of the correlation function
in question by means of the Landau equations LDL . (An illustration of their
application may be found in the Appendix of Ref. ESRr .)

##  3 Tetraquark Mesons: Genuinely Flavour-Exotic Quark Composition

In order to keep all following applications of our notion of multiquark
adequacy as transparent as possible, let us focus to the _flavour-exotic_
tetraquarks: bound states of four (anti-) quarks of mutually different quark
flavours, specified by further restriction of the characterization (2) by

$T=[\overline{q}_{a}\,q_{b}\,\overline{q}_{c}\,q_{d}]\ ,\qquad
a,b,c,d\in\\{u,d,s,c,b\\}\ ,\qquad a\neq b\neq c\neq d\ .$ (10)

Resorting to our criterion of Sect. 2, we identify QCD-level contributions to
the four-point correlation functions (5) maybe decisive for tetraquarks (10)
by sorting out any irrelevant one. For these, the configuration-space
contractions (6) and (7) provide the sought tetraquark-phile two-point and
three-point correlation functions; pairing any such correlation function with
the appropriate hadron-level counterpart then yields the corresponding QCD sum
rule, cf. Sect. 4.

For (at least) flavour-exotic tetraquarks, the correlation functions (5) may
be discriminated according to whether quark-flavour distributions of incoming
and outgoing states are identical (Subsect. 3.1) or different (Subsect. 3.2);
we tag the latter possibilities “flavour-preserving” or “flavour-retaining”
and “flavour-rearranging” or “flavour-reordering” or the like, respectively.

###  3.1 Flavour-Preserving Four-Point Correlation Functions of Interpolating
Currents

With respect to the complexity of the analysis, this task is definitely easier
if the distribution of the quark flavours is identical in initial and final
state of the four-point correlation function (5),

$\left\langle{\rm
T}\\!\left(j_{\overline{a}b}(y)\,j_{\overline{c}d}(y^{\prime})\,j^{\dagger}_{\overline{a}b}(x)\,j^{\dagger}_{\overline{c}d}(x^{\prime})\right)\right\rangle,\qquad\left\langle{\rm
T}\\!\left(j_{\overline{a}d}(y)\,j_{\overline{c}b}(y^{\prime})\,j^{\dagger}_{\overline{a}d}(x)\,j^{\dagger}_{\overline{c}b}(x^{\prime})\right)\right\rangle.$
(11)

####  3.1.1 Two-Point Correlation Functions: Two Identical Tetraquark
Interpolating Operators

The analysis of the flavour-preserving case is comparatively easy ESRp :
Figure 1 exemplifies the lower-order perturbative QCD-level contributions to
the four-point correlation functions (11). It poses no problem to convince
oneself that any tetraquark-phile among these features at least two gluon
exchanges exhibiting appropriate topology, like the one depicted in Fig. 1(c).
Then, as indicated by Fig. 2, the contraction (6) of a correlation function
(11) decomposes into a pair of ordinary-meson QCD sum rules (Fig. 3), and a
tetraquark-friendlier QCD sum rule (Fig. 4).

Figure 1: Flavour-_preserving_ correlation function ESRp ; TMA2 :
representative examples of contributions of the three (a,b,c) lowest
perturbative orders and their contractions to two tetraquark-interpolating
operators $\theta$. Figure 2: Flavour-_preserving_ two-tetraquark correlation
function ESRp ; TMA2 , disintegrable into (top row) two _ordinary_ -meson QCD
sum rules and (bottom row) a QCD sum rule possibly supporting tetraquark
poles. Figure 3: Conventional QCD sum rule QSR  preferably applicable to
_ordinary_ mesons (blue dashed lines). Figure 4: Refined QCD sum rule ESRp
likely improving analyses of _tetraquarks_ (blue dashed double line).

Phrased with a little bit sense of humour, the outcomes of the aforegoing
discussion for the flavour-preserving partition can be subsumed by the kind of
“graphical-mathematics” relation

$\mbox{Fig.\leavevmode\nobreak\
\ref{Fsg}}=2\times(\mbox{Fig.\leavevmode\nobreak\
\ref{Fso}})+\mbox{Fig.\leavevmode\nobreak\ \ref{Fst}}\ .$ (12)

####  3.1.2 Three-Point Correlation Function – One Tetraquark and Two
Conventional Mesons

Given the insights established generally for the four-point correlation
functions (11), identical conclusions about nature of tetraquark-phile
contributions (Fig. 5), decomposition of deduced quark–hadron relations and
tetraquark-fitting QCD sum rules will hold for the contraction (7).

Figure 5: Flavour-_retaining_ correlation function ESRp : typical lowest-order
_tetraquark-phile_ contribution, subject to a single contraction to a
tetraquark-interpolating operator, leaving two quark-bilinear currents.

###  3.2 Flavour-Regrouping Four-Point Correlation Function of Interpolating
Currents

Easy to guess, we now turn to the only other option for quark-flavour
distribution in four-point correlation functions (5), viz., that with
different flavour arrangement in initial and final states:

$\left\langle{\rm
T}\\!\left(j_{\overline{a}b}(y)\,j_{\overline{c}d}(y^{\prime})\,j^{\dagger}_{\overline{a}d}(x)\,j^{\dagger}_{\overline{c}b}(x^{\prime})\right)\right\rangle.$
(13)

Figure 6: Flavour-_redistributing_ correlation function ESRp ; ESRr :
illustrative examples of contributions of the three (a,b,c) lowest
perturbative orders and their contractions to two tetraquark-interpolating
operators $\theta$.

####  3.2.1 Two-Point Correlation Functions: Two Different Tetraquark
Interpolating Operators

In the flavour-reordering case ESRp ; ESRr , things get slightly more
complicated by the fact that none of the QCD-level contributions (Fig. 6)
proves to be configuration-space separable. In order to figure out which of
these are tetraquark-phile, we indeed have to exploit the Landau equations LDL
. These again point to the necessity of two or more gluon exchanges of
adequate topology, thus opening the path for a systematics of contributions
relevant or not for tetraquarks (Fig. 7).

####  3.2.2 Three-Point Correlation Function – One Tetraquark and Two
Conventional Mesons

Due to their common origin in the four-point correlation function (5) already
noted, the results gained for the two-point contraction evidently hold for the
three-point contraction (Fig. 8) too.

Figure 7: Flavour-_reordering_ correlation function of four quark-bilinear
operators ESRr , disintegrable into (a) an equation with no two-meson
$s$-channel cut (hatched rectangle) and (b) a tetraquark-adequate QCD sum rule
with two-meson $s$-channel cut (filled rectangle) and even tetraquark poles
(blue horizontal bar). Figure 8: Flavour-_rearranging_ correlation function
ESRr : lowest-order _tetraquark-phile_ contribution only once contracted to
one tetraquark-interpolating operator, leaving two quark-bilinear currents
untouched.

##  4 Tetraquark-Adequate QCD Sum Rules: Flavour-Exotic Tetraquarks

Application of the prescription, indicated in Sect. 1, for the extraction of a
QCD sum rule from correlation functions of hadron interpolating operators
entails a kind of “canonical” _structure_ :

Evaluation of the correlation function at QCD level necessitates two sorts of
contributions.

  * •

On the one hand, there exist perturbative contributions, discriminated by the
involvement of the strong coupling and usually cast in the form of a
dispersion integral of a spectral density. In this integration, the assumed
cancellation of higher QCD vs. higher hadron contributions is taken into
account by an effective threshold $s_{\rm eff}$ which by more in-depth
analysis has been revealed to depend (in a well-defined manner) on the Borel
parameter $\tau$: $s_{\rm eff}=s_{\rm eff}(\tau)$ ET1 ; ET2 ; ET3 .

  * •

On the other hand, there are nonperturbative contributions, favourably
subsumed in vacuum condensates – vacuum expectation values of products of the
quark and gluon field operators of QCD – which may be understood as a kind of
effective parameters of QCD. Upon a Borel transformation, that is, in their
Borel-transformed (or “Borelized”, for short) form, these get multiplied by
powers of its Borel parameter, whence they are also called _power
corrections_.

For the set of flavour-exotic tetraquarks, the lower-order perturbative QCD-
level contributions have been analyzed in Sect. 3; an analogous – albeit
somewhat more involved – discussion can be carried out, with similar results,
for the nonperturbative QCD-level contributions ESRp ; ESRr ; LMS10 .

Evaluation of this correlation function at the level of the spectrum of hadron
states leads to expressions that involve – in addition to the Borel parameter
$\tau$ – (some of) the hadron-relevant quantities of desire. In the case of
any flavour-exotic tetraquark state $|T\rangle$ specified by Eq. (10), these
observables include its mass $M$, its decay constants $f$, and its
vacuum–tetraquark matrix elements of a pair of quark–antiquark bilinear
operators (4) in momentum space, as defined by

$f_{\overline{a}b\overline{c}d}\equiv\langle
0|\theta_{\overline{a}b\overline{c}d}|T\rangle\ ,\qquad
f_{\overline{a}d\overline{c}b}\equiv\langle
0|\theta_{\overline{a}d\overline{c}b}|T\rangle\ ,$ (14)
$\begin{array}[]{l}\langle 0|{\rm
T}[j_{\overline{a}b}(y)\,j_{\overline{c}d}(y^{\prime})]|T\rangle\xrightarrow[\text{transformation}]{\text{Fourier}}A(T\to
j_{\overline{a}b}\,j_{\overline{c}d})\ ,\\\\[8.61108pt] \langle 0|{\rm
T}[j_{\overline{a}d}(y)\,j_{\overline{c}b}(y^{\prime})]|T\rangle\xrightarrow[\text{transformation}]{\text{Fourier}}A(T\to
j_{\overline{a}d}\,j_{\overline{c}b})\ .\end{array}$ (15)

By equating the outcomes of the above two levels of evaluation, we eventually
arrive at the generic shape of multiquark-adequate QCD sum rules. It reads,
for flavour-exotic tetraquarks,

$\displaystyle(f_{\overline{a}b\overline{c}d})^{2}\exp(-M^{2}\,\tau)$
$\displaystyle=\int_{\hat{s}}^{s_{\rm eff}(\tau)}{\rm
d}s\exp(-s\,\tau)\,\rho_{\rm p}(s)+\mbox{Borelized power corrections}\ ,$
$\displaystyle f_{\overline{a}b\overline{c}d}\,A(T\to
j_{\overline{a}b}\,j_{\overline{c}d})\exp(-M^{2}\,\tau)$
$\displaystyle=\int_{\hat{s}}^{s_{\rm eff}(\tau)}{\rm
d}s\exp(-s\,\tau)\,\Delta_{\rm p}(s)+\mbox{Borelized power corrections}\ ,$
$\displaystyle
f_{\overline{a}b\overline{c}d}\,f_{\overline{a}d\overline{c}b}\exp(-M^{2}\,\tau)$
$\displaystyle=\int_{\hat{s}}^{s_{\rm eff}(\tau)}{\rm
d}s\exp(-s\,\tau)\,\rho_{\rm r}(s)+\mbox{Borelized power corrections}\ ,$
$\displaystyle f_{\overline{a}d\overline{c}b}\,A(T\to
j_{\overline{a}b}\,j_{\overline{c}d})\exp(-M^{2}\,\tau)$
$\displaystyle=\int_{\hat{s}}^{s_{\rm eff}(\tau)}{\rm
d}s\exp(-s\,\tau)\,\Delta_{\rm r}(s)+\mbox{Borelized power corrections}\ ,$

wherein the $s$-dependent functions $\rho_{\rm p,r}$ and $\Delta_{\rm p,r}$
denote, respectively, the flavour-preserving (p) and flavour-rearranging (r)
spectral densities in their tetraquark-phile refinement explicated in Sect. 2,
all originating in the two-point ($\rho$) and three-point ($\Delta$)
correlation functions of Sect. 3.

Acknowledgements. Both D. M. and H. S. would like to thank for support by
joint CNRS/RFBR Grant PRC Russia/19-52-15022, D. M. for support by Austrian
Science Fund (FWF) Project P29028-N27, and H. S. for support by EU research
and innovation program Horizon 2020 under Grant Agreement 824093.

## References

  * (1) M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147, 385 (1979)
  * (2) K.G. Wilson, Phys. Rev. 179, 1499 (1969)
  * (3) W. Lucha, D. Melikhov, S. Simula, Phys. Rev. D 79, 096011 (2009)
  * (4) W. Lucha, D. Melikhov, S. Simula, J. Phys. G 37, 035003 (2010)
  * (5) W. Lucha, D. Melikhov, S. Simula, Phys. Lett. B 687, 48 (2010)
  * (6) W. Lucha, D. Melikhov, H. Sazdjian, Phys. Rev. D 100, 014010 (2019)
  * (7) W. Lucha, D. Melikhov, H. Sazdjian, Phys. Rev. D 100, 074029 (2019)
  * (8) W. Lucha, D. Melikhov, H. Sazdjian, PoS (EPS-HEP2019), 536 (2020)
  * (9) W. Lucha, D. Melikhov, H. Sazdjian, EPJ Web Conf. 222, 03016 (2019)
  * (10) W. Lucha, D. Melikhov, H. Sazdjian, Phys. Rev. D 103, 014012 (2021)
  * (11) W. Lucha, D. Melikhov, H. Sazdjian, Supl. Rev. Mex. Fís. 3, 0308035 (2022)
  * (12) W. Lucha, D. Melikhov, H. Sazdjian, EPJ Web Conf. 192, 00044 (2018)
  * (13) R.L. Workman et al. (Particle Data Group), PTEP 2022, 083C01 (2022)
  * (14) R.L. Jaffe, Nucl. Phys. A 804, 25 (2008)
  * (15) M. Fierz, Z. Phys. 104, 553 (1937)
  * (16) W. Lucha, D. Melikhov, H. Sazdjian, Prog. Part. Nucl. Phys. 120, 103867 (2021)
  * (17) W. Lucha, D. Melikhov, H. Sazdjian, Phys. Rev. D 96, 014022 (2017)
  * (18) W. Lucha, D. Melikhov, H. Sazdjian, Eur. Phys. J. C 77, 866 (2017)
  * (19) W. Lucha, D. Melikhov, H. Sazdjian, PoS (EPS-HEP 2017), 390 (2018)
  * (20) W. Lucha, D. Melikhov, H. Sazdjian, Phys. Rev. D 98, 094011 (2018)
  * (21) L.D. Landau, Nucl. Phys. 13, 181 (1959)