# Is the LHCb $P_{c}(4312)^{+}$ plausible in the GlueX $\gamma p\to J/\psi p$
total cross sections ?

Igor Strakovsky Corresponding author<EMAIL_ADDRESS>Institute for Nuclear
Studies, Department of Physics, The George Washington University, Washington,
DC 20052, USA  William J. Briscoe Institute for Nuclear Studies, Department of
Physics, The George Washington University, Washington, DC 20052, USA  Eugene
Chudakov Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA  Ilya Larin University of Massachusetts (UMASS Amherst),
Amherst, MA 01003, USA  National Research Centre “Kurchatov Institute,” Moscow
117218, Russia  Lubomir Pentchev Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606, USA  Axel Schmidt Institute for
Nuclear Studies, Department of Physics, The George Washington University,
Washington, DC 20052, USA  Ronald L. Workman Institute for Nuclear Studies,
Department of Physics, The George Washington University, Washington, DC 20052,
USA

###### Abstract

New high-statistics total cross section data for $\gamma p\to J/\psi p$ from
the GLUonic EXcitation (GlueX) experiment are fitted in a search for the
exotic $P_{c}(4312)^{+}$ state observed by the Large Hadron Collider beauty
(LHCb) collaboration. The integrated luminosity of this GlueX experiment was
about $320~{}\mathrm{pb^{-1}}$. The fits show that destructive interference
involving an $S$-wave resonance and associated non-resonance background
produces a sharp dip structure about $75~{}\mathrm{MeV}$ below the LHCb mass,
in the same location as a similar structure is seen in the data. Limitations
of the employed model and the need for improved statistics are discussed.

The formation of hadrons from quarks and gluons is governed by quantum
chromodynamics (QCD). The traditional quark model Gell-Mann:1964ewy ;
Zweig:1964jf , where hadrons are classified as mesons (composed of
$q\overline{q}$) and baryons (composed of $qqq$), has provided a satisfactory
description of the hadrons observed over the last 75 years. However, the last
two decades have witnessed the emergence of states or resonant structures in
experiments, many of which do not fit the hadron spectrum predicted by the
naive quark model and thus are candidates to be exotic states. In particular,
the Large Hadron Collider beauty (LHCb) experiment has reported the
observation of four narrow $P_{c}^{+}$ pentaquark states in decays

$\Lambda_{b}^{0}\to J/\psi+p+K^{-}\>,$ $\Lambda_{b}^{0}\to
J/\psi+p+\pi^{-}\>,$

and

$B_{s}^{0}\to J/\psi+p+\overline{p}$

with hidden heavy charm quark contributions Aaij:2015tga ; LHCb:2016lve ;
Aaij:2019vzc ; LHCb:2021chn . The masses of these four narrow $P_{c}$ states,
as well as their significance, are given in Table 1. In response, many
theoretical analyses evaluated these cases (see for instance, Refs.
Wang:2015jsa ; Kubarovsky:2015aaa ; Karliner:2015voa ; Blin:2016dlf ;
Eides:2019tgv ; Semenova:2019gzf ; Burns:2021jlu ; Peng:2019wys ;
Zhang:2023czx ). In particular, in Refs. Wang:2015jsa ; Kubarovsky:2015aaa ;
Karliner:2015voa ; Blin:2016dlf , it was proposed to search for this $P_{c}$
state in the $J/\psi$ photoproduction off the nucleon. The corresponding cross
section was evaluated based on the Vector Meson Dominance (VMD) model Gell-
Mann:1961jim ; Kroll:1967it ; Sakurai:1969 .

Table 1: Summary of $P_{c}^{+}$ properties reported by the LHCb Collaboration Aaij:2019vzc ; LHCb:2021chn .  State | M | $\Gamma[Pc\to J/\psi+p]$ | Significance
---|---|---|---
| (MeV) | (MeV) |
$P_{c}(4312)^{+}$ | $4311.9\pm 0.7_{-0.6}^{+6.8}$ | $9.8\pm 2.7_{-4.5}^{+3.7}$ | 7.3 $\sigma$
$P_{c}(4337)^{+}$ | $4337{{}^{+7}_{-4}}{{}^{+2}_{-2}}$ | $29{{}^{+26}_{-12}}{{}^{+13}_{-14}}$ | 3.1 – 3.7 $\sigma$
$P_{c}(4440)^{+}$ | $4440.3\pm 1.3_{-4.7}^{+4.1}$ | $20.6\pm 4.9_{-10.1}^{+8.7}$ | 5.4 $\sigma$
$P_{c}(4457)^{+}$ | $4457.3\pm 0.6_{-1.7}^{+4.1}$ | $6.4\pm 2.0_{-1.9}^{+5.7}$ | 5.4 $\sigma$

The GLUonic EXcitation (GlueX) experiment measured total cross sections for
the reaction $\gamma+p\to J/\psi+p$ and reported no evidence for the LHCb
pentaquark candidates $P_{c}^{+}$ (Fig. 1) and set model-dependent upper
limits on their branching ratios (BRs) Ali:2019lzf .

These interpretations of the LHCb and GlueX cases considered a constructive
interference between a resonance and non-resonant background for particular
wave, $L^{J}$ (here, $L$ is a total orbital momentum and $J$ is the total
angular momentum). In this work, we relax this requirement and consider the
possibility of destructive interference as well.

Figure 1: $s$-channel resonance $P_{c}$.

New higher-statistics GlueX measurements of the threshold total cross section,
for the reaction $\gamma+p\to J/\psi+p$ GlueX:2023 , motivate an alternative
search for the LHCb exotics. The integrated luminosity of this recent GlueX
experiment was about $320~{}\mathrm{pb^{-1}}$, a large improvement over the
approximately $68~{}\mathrm{pb^{-1}}$ of the GlueX data presented in Ref.
Ali:2019lzf ). Within the limits of the available statistics, the clearest
structure is a dip observed in the recent GlueX total $\gamma+p\to J/\psi+p$
cross sections, in vicinity of $W=4.3~{}\mathrm{GeV}$ GlueX:2023  (Fig. 2), a
possible indication of the lowest-lying LHCb $P_{c}(4312)$ Aaij:2019vzc . This
dip becomes more pronounced in new GlueX data GlueX:2023  versus the initial
GlueX report Ali:2019lzf . We therefore consider whether simple
background/resonance interference is capable of reproducing the low-energy
behavior visible in the cross section data. Through interference, the cross
section can develop either a peak or a dip. Pedagogical reviews of quantum
interference are given in Refs. Azimov:2009ta ; LeviSetti:1973 .

A general expressions for the total cross section of the inelastic binary
reaction, $\gamma+p\to J/\psi+p$, can be written as

$\sigma_{t}=\int_{0}^{2\pi}\int_{0}^{\pi}\frac{d\sigma}{d\Omega}~{}\sin\theta~{}d\theta~{}d\phi\>,$
(1)

where $\theta$ and $\phi$ are the $J/\psi$ polar and azimuthal production
angles, respectively.

Phenomenologically, the total cross section ($\sigma_{t}$), using the Landau-
Lifshitz normalisation, Landau:1991wop  is

$\sigma_{t}=\frac{\pi}{4k^{2}}\sum_{J=0}^{\infty}(2J+1)~{}|f|^{2}\>,$ (2)

where $k$ is the photon momentum in the $\gamma p$ center-of-mass (c.m.)
system.

For an $s$-channel resonance in the reaction $\gamma+p\to P_{c}^{+}\to
J/\psi+p$, in the considered partial wave, $L^{J}$,(Fig. 1), we shall separate
the non-resonant ($b$) and resonance ($R$) parts in the partial amplitude
($f$) following Ref. Strakovsky:1984hb ; LeviSetti:1973 . Then for the
dominant partial wave, we can write

$f=b+R\cdot\exp(2i\alpha)\>,$ (3)

where $\alpha$ is the relative phase shift which is responsible for the
interference between resonance ($R$) and non-resonant ($b$) components of the
partial amplitude. The interference may be either positive (constructive) or
negative (destructive). We make $\alpha$ a free parameter in our fit of the
cross sections, not given by any theory. Let us note that the resonance can
interfere differently in different decay channels, at least due to different
properties of the corresponding backgrounds.

For the resonance part of the partial amplitude, $f$, we shall use a canonical
relativistic Breit-Wigner parameterization:

$R=\frac{2\Gamma M}{[(M)^{2}-s]-i\Gamma M}~{}X~{}\>,$ (4)

where $s$ is the square of the total energy in c.m. Then, $M$, $\Gamma$, and
$X$ are respectively the mass and total and partial widths

$X=\frac{\sqrt{\Gamma(\gamma+p)~{}\Gamma(J/\psi+p)}}{\Gamma}$
$~{}~{}~{}~{}=\sqrt{X(\gamma+p)~{}X(J/\psi+p)}\>$ (5)

of a particular resonance. Here, $\Gamma(\gamma+p)$ and $\Gamma(J/\psi+p)$ are
the partial decay widths of the resonance $P_{c}$ in to two-particle channels,
$P_{c}\to\gamma+p$ and $P_{c}\to J/\psi+p$, respectively. Since the width
$\Gamma$ of $P_{c}$ is rather small, we shall assume a width with no energy
dependence avoiding the need to modify the Breit-Wigner parameterization with
additional form factors, leaving just three free resonance parameters $M$,
$\Gamma$, and $X$ to fit.

Traditionally, the total cross section behavior vs energy of the near-
threshold binary inelastic reaction $m_{a}+m_{b}<m_{c}+m_{d}$ is described as
series of odd powers in $q$ (even powers in the case of elastic scattering),
as we used for the $J/\psi-p$ scattering length determination recently
Strakovsky:2019bev :

$\sigma_{t}=A~{}q+B~{}q^{3}+C~{}q^{5}\>,$ (6)

where $A$, $B$, and $C$ are free parameters and $q$ is the c.m. momentum of
the vector meson in the $J/\psi-p$ system. The linear term is determined by
two independent $S$-waves only with the total spin 1/2 and/or 3/2.
Contribution to the cubic term comes from both $P$-wave amplitudes and energy
dependence of $S$-wave amplitudes. Then, the fifth-order term arises from
$D$-waves and energy dependencies of $S$\- and $P$-waves. The phenomenological
study of the $J/\psi-p$ scattering length, using GlueX data Ali:2019lzf , has
shown that parameters $B$ and $C$ parameters are consistent with zero (within
uncertainties) Strakovsky:2019bev . For that reason, for the background (non-
resonant) amplitude, we will use just a linear and cubic terms from Eq. (6):

$b=\sqrt{A~{}q~{}+~{}B~{}q^{3}}\>,$ (7)

In general, three other (higher mass) $P_{c}$s should reveal themselves in the
$\gamma+p\to J/\psi+p$ cross section. This would require better statistics and
a finer energy binning. If the GlueX Collaboration treats the two total cross
section points in the vicinity of $W=4.3~{}\mathrm{GeV}$ as a potential dip
(taking into account statistical and point-to-point systematic uncertainties),
the probability that they are not a statistical fluctuation from an underlying
smooth fit to the observed cross sections corresponds to a significance of
$2.6~{}\mathrm{\sigma}$ GlueX:2023 . However, if one considers the probability
for any two adjacent points in the whole energy interval
($W=4.05-4.71~{}\mathrm{GeV}$) to have a deviation of at least this size, the
significance reduces to $1.4~{}\mathrm{\sigma}$ GlueX:2023 .

Figure 2: Best-fit results for new GlueX total cross sections for the reaction
$\gamma+p\to J/\psi+p$ (blue filled circles) GlueX:2023 . The vertical error
bars represent the total uncertainties (statistical and point-to-point
systematic uncertainties in quadrature). The horizontal error bars reflect the
energy binning (not used in the fit). The best-fit result, using Eq. (3),
shown by red solid curve. Green dashed curve corresponds to the non-resonant
fit as a function of $q$ (Eq. (7)). Blue dash-dotted curve corresponds to the
$S$-wave resonance. The red vertical arrow indicates the $J/\psi$ production
threshold ($W=4.035~{}\mathrm{GeV}$).

Because the current statistics of the GlueX measurements $2270\pm 58$ data
GlueX:2023  is by a factor of about 5 higher than recently published $469\pm
22$ Ali:2019lzf , let us look for the effect of the LHCb low-lying $P_{c}$
through the destructive interference of the $S$-wave resonance and associated
non-resonance background.

Table 2: Best-fit results for the new GlueX data using Eq. (2). 2nd column presents results for statistical and point-to-point systematic uncertainties which are combined in quadrature.  Quantity | Units | Parameter
---|---|---
M | $\mathrm{MeV}$ | 4235$\pm$ 8
$\Gamma$ | $\mathrm{MeV}$ | 35.4$\pm$ 8.2
X |  | 0.023$\pm$ 0.005
$\alpha$ | $\mathrm{deg}$ | 40.8$\pm$ 5.7
A | $\mathrm{nb\cdot GeV/c}$ | 0.00251$\pm$ 0.00046
B | $\mathrm{nb/(GeV/c)}$ | 0.00688$\pm$ 0.00083

The best-fit results for new GlueX total cross sections for the reaction
$\gamma+p\to J/\psi+p$ GlueX:2023 , using Eq. (2) is given in Table 2 and is
shown on Figs. 2 and 3. Additionally, we found a sensitivity to the relative
phase shift, $\alpha$.

Excluding a resonance contribution from Eq. (3), one can get the following
result for the non-resonant contribution, using Eq. (7):
$A=(0.00183\pm 0.00040)~{}\mathrm{nb\cdot GeV/c}$ ,
$B=(0.00766\pm 0.00077)~{}\mathrm{nb/(GeV/c)}$ ,
There is reasonable agreement between non-resonant parameters of two fits.

There were 6 free parameters for the $P_{c}(4312)^{+}$ resonance in the
$S$-wave case and the overall $\mathrm{\chi^{2}/ndf}$ = 11.99/12 = 1.00. The
best-fit for the alternative hypothesis (just non-resonant case) gives
$\mathrm{\chi^{2}/ndf}$ = 19.74/16 = 1.23. The $A$ parameter is very small and
in agreement with a very small $J/\psi-p$ scattering length determined
recently Strakovsky:2019bev ; Pentchev:2020kao ; Du:2020bqj . Overall, our
phenomenology does not allow us to select a particular partial wave for the
$P_{c}(4312)^{+}$ resonance ($S$\- or $P$-wave). To get a $P$-wave
$P_{c}(4312)$ resonance partial width, one needs $X/\sqrt{3}$ and $A/\sqrt{3}$
with $B/\sqrt{3}$.

Here we have shown that a resonance-like structure is “plausiable” in the
GlueX data GlueX:2023 , in an energy region close to the low-mass LHCb
pentaquark Aaij:2019vzc ; LHCb:2021chn . The shift in masses between GlueX and
LHCb results ($77~{}\mathrm{MeV}$) may depend on the reaction mechanism
(including cusps (open charm) and background choices). One should note that if
a “bump” is imposed on the GlueX data “by hand” (consider the 7th - 9th energy
values up from threshold), a qualitative description of the data up to
$4.35~{}\mathrm{GeV}$ is possible, but with a higher chi-squared, if the above
fit form is used. The statistics of the present GlueX data shown in Figs. 2
and 3 is not sufficient to draw a definite conclusion.

Figure 3: The GlueX total cross section for the $\gamma+p\to J/\psi+p$
photoproduction (blue filed circles) GlueX:2023 . The open charm model
predictions Du:2020bqj  shown by magenta dashed (green dash-dotted) curves
with $q_{max}$ = 1 GeV/$c$ ($q_{max}$ = 1.2 GeV/$c$). This model does not fit
the GlueX data and has no normalization factor. The phenomenological best-fit
result using Eq. (3) shown by red solid curve. Vertical black dotted lines
show $\Lambda_{c}\bar{D}^{(\ast)}$-thresholds. The red vertical arrow
indicates the $J/\psi$ production threshold ($W=4.035~{}\mathrm{GeV}$).

We conclude with an admission that the analytical structure in the vicinity of
the LHCb $P_{c}$ resonances is not as simple as a resonance plus background
treatment would suggest. There are additional contributions to the $P_{c}$
effect associated with the open charm impact Du:2020bqj . In particular, the
$\Lambda_{c}\bar{D}^{(\ast)}$ – cusp effects could be visible. Actually, the
interference between open charm and gluon exchange may (by some accident)
produce a dip (see Fig. 3) but there is room for the resonance. The green
dash-dotted curve ($q_{max}$ = 1.2 GeV/$c$) reasonably agrees with the new
GlueX data. While not evident in the GlueX data, one cannot exclude that we
have all 4 LHCb $P_{c}$ resonances Aaij:2019vzc  together with open charm and
gluon exchange (gluon contribution can be strongly suppressed due to the
“young” effect Feinberg:1980yu , details about “young” effect in the vector
meson and nucleon interaction at the threshold is given in Ref.
Strakovsky:2021vyk ). Complicated interference between these different
components should motivate new high statistics experiments.

We thank Misha Ryskin, Misha Voloshin, Anatoly Dolgolenko, Moskov Amaryan, and
Michael Eides for useful remarks and continuous interest in the paper and
Feng-Kun Guo with Meng-Lin Du for $J/\psi p$ cross section predictions. This
work was supported in part by the U. S. Department of Energy, Office of
Science, Office of Nuclear Physics, under Awards No. DE–SC0016583 and
DE–SC0016582 and Contract No. DE-–AC05-–06OR23177.

## References

  * (1) M. Gell-Mann, “A schematic model of baryons and mesons,” Phys. Lett. 8, 214 (1964).
  * (2) G. Zweig, “An SU(3) model for strong interaction symmetry and its breaking,” Preprint CERN–TH–412 (1964).
  * (3) R. Aaij et al. [LHCb Collaboration], “Observation of $J/\psi p$ resonances consistent with pentaquark states in $\Lambda_{b}^{0}\to J/\psi K^{-}p$ decays,” Phys. Rev. Lett. 115, 072001 (2015).
  * (4) R. Aaij et al. [LHCb Collaboration], “Evidence for exotic hadron contributions to $\Lambda_{b}^{0}\to J/\psi p\pi^{-}$ decays,” Phys. Rev. Lett. 117, 082003 (2016).
  * (5) R. Aaij et al. [LHCb Collaboration], “Observation of a narrow pentaquark state, $P_{c}(4312)^{+}$, and of two-peak structure of the $P_{c}(4450)^{+}$,” Phys. Rev. Lett. 122, 222001 (2019).
  * (6) R. Aaij et al. [LHCb Collaboration], “Evidence for a new structure in the $J/\psi p$ and $J/\psi\bar{p}$ systems in $B_{s}^{0}\to J/\psi p\bar{p}$ decays,” Phys. Rev. Lett. 128, 062001 (2022).
  * (7) Q. Wang, X. H. Liu, and Q. Zhao, “Photoproduction of hidden charm pentaquark states $P_{c}^{+}(4380)$ and $P_{c}^{+}(4450)$,” Phys. Rev. D 92, 034022 (2015).
  * (8) V. Kubarovsky and M. B. Voloshin, “Formation of hidden-charm pentaquarks in photon-nucleon collisions,” Phys. Rev. D 92, 031502 (2015).
  * (9) M. Karliner and J. L. Rosner, “Photoproduction of exotic baryon resonances,” Phys. Lett. B 752, 329 (2016).
  * (10) A. N. Hiller Blin, C. Fernández-Ramírez, A. Jackura, V. Mathieu, V. I. Mokeev, A. Pilloni, and A. P. Szczepaniak, “Studying the Pc(4450) resonance in J/$\psi$ photoproduction off protons,” Phys. Rev. D 94, 034002 (2016).
  * (11) M. I. Eides, V. Y. Petrov, and M. V. Polyakov, “New LHCb pentaquarks as hadrocharmonium states,” Mod. Phys. Lett. A 35, 2050151 (2020).
  * (12) A. N. Semenova, V. V. Anisovich, and A. V. Sarantsev, “New narrow LHCb pentaquarks as lowest antiquark-diquark-diquark systems,” Eur. Phys. J. A 56, 142 (2020).
  * (13) T. J. Burns and E. S. Swanson, “Experimental constraints on the properties of $P_{c}$ states,” Eur. Phys. J. A 58, 68 (2022).
  * (14) F. Z. Peng, M. Z. Liu, Y. W. Pan, M. Sánchez Sánchez, and M. Pavon Valderrama, “Five-flavor pentaquarks and other light- and heavy-flavor symmetry partners of the LHCb hidden-charm pentaquarks,” Nucl. Phys. B 983, 115936 (2022).
  * (15) Z. Zhang, J. Liu, J. Hu, Q. Wang, and U. G. Meißner, “Revealing the nature of hidden charm pentaquarks with machine learning,” [arXiv:2301.05364 [hep-ph]].
  * (16) M. Gell-Mann and F. Zachariasen, “Form-factors and vector mesons,” Phys. Rev. 124, 953 (1961).
  * (17) N. M. Kroll, T. D. Lee, and B. Zumino, “Neutral vector mesons and the hadronic electromagnetic current,” Phys. Rev. 157, 1376 (1967).
  * (18) J. J. Sakurai, “Currents and mesons,” (The University of Chicago Press, 1969).
  * (19) A. Ali et al. [GlueX Collaboration], “First measurement of near-threshold $J/\psi$ exclusive photoproduction off the proton,” Phys. Rev. Lett. 123, 072001 (2019).
  * (20) S. Adhikari et al. [GlueX Collaboration], “Measurement of the $J/\psi$ photoproduction cross section over the full near-threshold kinematic region,” arXiv:2304.03845 [nucl-ex],
  * (21) Y. Azimov, “Quantum interference of particles and resonances,” J. Phys. G 37, 023001 (2010).
  * (22) R. Levi Setti and T. Lasinski, “Strongly Interacting Particles (Chicago lectures in physics),” (The University of Chicago Press, 1973).
  * (23) L. D. Landau and E. M. Lifshitz, “Quantum Mechanics : Non-Relativistic Theory,” (Butterworth-Heinemann, 1977).
  * (24) I. I. Strakovsky, A. V. Kravtsov, and M. G. Ryskin, “The study of the $\pi^{+}d\to pp$ process within the framework of $J\leq 4$ phase shift analysis,” Yad. Fiz. 40, 429 (1984) [Sov. J. Nucl. Phys. 40, 273 (1984)].
  * (25) M. L. Du, V. Baru, F. K. Guo, C. Hanhart, U. G. Meißner, A. Nefediev, and I. Strakovsky, “Deciphering the mechanism of near-threshold $J/\psi$ photoproduction,” Eur. Phys. J. C 80, 1053 (2020).
  * (26) I. Strakovsky, D. Epifanov, and L. Pentchev, “J/$\psi$p scattering length from GlueX threshold measurements,” Phys. Rev. C 101, 042201 (2020).
  * (27) L. Pentchev and I. I. Strakovsky, “$J/\psi$-$p$ scattering length from the total and differential photoproduction cross sections,” Eur. Phys. J. A 57, 56 (2021).
  * (28) E. L. Feinberg, “Hadron clusters and half dressed particles in Quantum Field Theory,” Usp. Fiz. Nauk 132, 225 (1980) [Sov. Phys. Usp. 23, 629 (1980)].
  * (29) I. I. Strakovsky, W. J. Briscoe, L. Pentchev, and A. Schmidt, “Threshold Upsilon-meson photoproduction at the EIC and EicC,” Phys. Rev. D 104, 074028 (2021).