Photo-absorption on deuteron contributed by d^*(2380) resonance

TL;DR

This paper calculates the photo-absorption cross section on deuteron due to the $d^*(2380)$ resonance, finding it significantly smaller than experimental data, and explores its structure as a six-quark system.

Contribution

It provides a theoretical calculation of the photo-absorption cross section involving the $d^*(2380)$ resonance considering its internal structure as a six-quark system.

Findings

Calculated cross section is about 10 nb, much smaller than experimental data.

Only next-to-leading order terms contribute significantly.

The model supports $d^*(2380)$ as a compact six-quark state.

Abstract

In order to understand the possible physical nature of the newly observed resonance $d^*(2380)$, we calculate the real photo-absorption cross section on deuteron contributed by the resonance $d^*(2380)$ by considering the electromagnetic transition amplitude of $\gamma +d\rightarrow d^*(2380)$. In our interpretation, the $d^*(2380)$ is regarded as a compact six-quark system with mainly two components of $\Delta\Delta$ and hidden-color clusters $C_8C_8$. We find that only the next-to-leading terms contribute the $\gamma +d\rightarrow d^*(2380)$ and the obtained photo-absorption cross section is quite small which is in the order of 10 $nb$. Compared with data measured at ELPH and Mainz recently, it is almost about 20 times smaller.

Tables

Table 1. Table 1: Isospin factor [ I I ​ n ​ t . ] ( a , b , c , d , e , f ) subscript delimited-[] superscript 𝐼 𝐼 𝑛 𝑡 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 \big{[}I^{Int.}\big{]}_{(a,b,c,d,e,f)} .

Intermediate state	${\left[I^{Int.}\right]}_{(a,b,c)}$	${\left[I^{Int.}\right]}_{(d,e,f)}$
$Int.=N,T^{Int.}=1/2$	$-\sqrt{2}$	$0$
$Int.=\Delta ,T^{Int.}=3/2$	$0$	$-\frac{\sqrt{3}}{2}$

Table 2. Table 2: Spin factor [ S I ​ n ​ t . ] ( a , b , c , d , e , f ) subscript delimited-[] superscript 𝑆 𝐼 𝑛 𝑡 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 \big{[}S^{Int.}\big{]}_{(a,b,c,d,e,f)} with B 2 ​ ( m d ) = C 1 , 1 ; 2 , m d 3 , ( m d + 1 ) ​ C 2 , 0 ; 2 , m d 1 , m d subscript 𝐵 2 subscript 𝑚 𝑑 superscript subscript 𝐶 1 1 2 subscript 𝑚 𝑑 3 subscript 𝑚 𝑑 1 superscript subscript 𝐶 2 0 2 subscript 𝑚 𝑑 1 subscript 𝑚 𝑑 B_{2}(m_{d})=C_{1,1;2,m_{d}}^{3,(m_{d}+1)}C_{2,0;2,m_{d}}^{1,m_{d}} and B 3 ​ ( m d ) = C 1 , 1 ; 3 , m d 3 , ( m d + 1 ) ​ C 2 , 0 ; 3 , m d 1 , m d subscript 𝐵 3 subscript 𝑚 𝑑 superscript subscript 𝐶 1 1 3 subscript 𝑚 𝑑 3 subscript 𝑚 𝑑 1 superscript subscript 𝐶 2 0 3 subscript 𝑚 𝑑 1 subscript 𝑚 𝑑 B_{3}(m_{d})=C_{1,1;3,m_{d}}^{3,(m_{d}+1)}C_{2,0;3,m_{d}}^{1,m_{d}} , respectively.

Intermediate state	${\left[S^{Int.}\left(m_d\right)\right]}_{(a,b,c)}$	${\left[S^{Int.}\left(m_d\right)\right]}_{(d,e,f)}$
$Int.=N,S^{Int.}=1/2$	$\sqrt{\frac{2}{9}}\left[B_2\left(m_d\right)-\sqrt{2}{B}_3\left(m_d\right)\right]$	$\frac{1}{2}\sqrt{\frac{10}{3}}{B}_2\left(m_d\right)$
$Int.=\Delta ,S^{int.}=3/2$	$\frac{\sqrt{5}}{6}\left[\sqrt{2}{B}_2\left(m_d\right)+\frac{8}{5}{B}_3\left(m_d\right)\right]$	$-\sqrt{\frac{28}{25}}{B}_3\left(m_d\right)$

Equations

$${\Psi }_{6q}$$

$$|d^{\ast }(S_{d^{\ast }}=3,m_{d^{\ast }})>$$

$$k^2=\frac{(Q^2+M_N^2+M_X^2{)}^2-4M_N^2{M}_X^2}{Q^2+2(M_N^2+M_X^2)},k_0=\sqrt{M_X^2+\frac{{\vec{k}}^2}{4}}-\sqrt{M_N^2+\frac{{\vec{k}}^2}{4}},$$

$${\mathcal{A}}_{(a,c,f)}^{Int.}(m_d)$$

$${\mathcal{D}}_a^{Int.}$$

$${\mathcal{D}}_c^{Int.}$$

$${\mathcal{D}}_f^{Int.}$$

$$\displaystyle\Big{[}{\cal C}_{a,c}^{Int.}\Big{]}$$

$${\mathcal{A}}_{(b,d,e)}^{Int.}(m_d)$$

$${\mathcal{D}}_b^{Int.}$$

$${\mathcal{D}}_d^{Int.}$$

$${\mathcal{D}}_e^{Int.}$$

$$\displaystyle\Big{[}{\cal C}_{b}^{Int.}\Big{]}$$

$$\mathcal{A}(m_d)=\sum _{i=a}^f\sum _{Int.=N}^{\Delta }{\mathcal{A}}_i^{Int.}(m_d).$$

$$\displaystyle\sigma^{res.}(W;m_{d})=\sum_{R}\frac{2M_{d}}{W+M_{R}}R(W,\Gamma)\big{|}{\cal A}(W;m_{d})\big{|}^{2},$$

$$R(W,\Gamma )=\frac{\Gamma }{(W-M_R{)}^2+{\Gamma }^2/4},$$

$$\displaystyle\sigma^{res.}(W;T)=\frac{1}{3}\Big{(}\sigma^{res.}(W;m_{d}=+1)+\sigma^{res.}(W;m_{d}=0)+\sigma^{res.}(W;m_{d}=-1)\Big{)}.$$

$$\displaystyle\sigma^{res.}(W=M_{R};T)=\frac{4}{3}\frac{M_{d}}{W_{R}\Gamma}\Big{(}|{\cal A}(M_{R};m_{d}=+1)|^{2}+|{\cal A}(M_{R};m_{d}=0)|^{2}+|{\cal A}(M_{R};m_{d}=-1)|^{2}\Big{)}.$$

Full text

Constituent quark model, $d^{*}(2380)$; photon-absorption cross section; deuteron

I Introduction

Since dibaryon states were proposed more than half century ago, their existence has become one of the most interest issues of hadronic physics. Among various dibaryon states, $H$ particle and $d^{*}$ were involved most. In particular, the $d^{*}$ state has been explicitly studied by different approaches from the hadronic degrees of freedom to the quark degrees of freedom, and the obtained binding energy was ranged from a few MeV to several hundred MeV. Searching for such an interesting state has also been considered as one of the aims in several experimental projects. However, no convincing results were released until 2009. After that, a series of experimental studies for $d^{*}$ was carried out in the analysis of ABC effect by CELSIUS/WASA and WASA@COSY Collaborations  CELSIUS-WASA ; Adlarson:2011bh ; Adlarson:2012fe ; Adlarson:2014pxj . Various double-pion and single-pion decays, including invariant mass spectra, Dalitz plots, Argon plots, in the $pn$ and $pA$ reactions, the analyzing power of the neutron-proton scattering and etc., have been measured and analyzed. It was found that the experimental results cannot be simply understood by the contribution either from the intermediate Roper excitation or from the t-channel $\Delta\Delta$ effect, except introducing an intermediate new resonance. Then, the discovery of a new resonance, with a mass (width) about $2370\sim 2380~{}\rm{MeV}$ ($70\sim 80~{}\rm{MeV}$) and the quantum numbers of $I(J^{P})=0(3^{+})$, was announced  CELSIUS-WASA ; Adlarson:2011bh ; Adlarson:2012fe ; Adlarson:2014pxj . It is believed that such a state is just the $d^{*}$ state which has been hunted for several decades due to its baryon number being 2. In general, it can be explained by either ”an exotic compact particle” or ”a hadronic molecule state” (see the review article of Ref. Clement:2016vnl ).

One may reasonably expect that the threshold (or cusp) effect may not be so significant in the $d^{*}$ case as in the XYZ particles due to the fact that the observed mass of $d^{*}$ is about $80~{}\rm{MeV}$ below the $\Delta\Delta$ threshold and about $70~{}\rm{MeV}$ above the $\Delta\pi N$ threshold Chen:2016qju ; Guo:2017jvc ; Dong:2017gaw . If $d^{*}$ does exist, it contains at least 6 light quarks, and it is also much different from the XYZ particles which contain heavy flavor.

Up to now, many theoretical models for the structure of $d^{*}$ have been developed or proposed. There are mainly two structural schemes which attract considerable attention of community. One assumes that the $d^{*}$ state has a compact structure, and may be an exotic hexaquark dominated state whose mass is about $2380-2414~{}\rm{MeV}$ and width about $71~{}\rm{MeV}$, respectively  Yuan ; Brodsky ; Huang:2014kja ; Huang:2015nja ; Dong ; Dong1 ; Dong2 ; Dong2017 . Some quark models calculations for the dibaryon $d^{*}(2380)$ are also referred to Ping:2008tp ; Huang:2013rla . The other one, in order to explain the upper limit of the single-pion decay width of $d^{*}$ Clement2017 , proposes that the $d^{*}$ state is basically a molecular-like hadronic state Gal1 , which originates from a three-body $\Delta N\pi$ resonance assumption, where the pole position of the resonance locates around ($2363\pm 20$) $+{\it i}(65\pm 17)$ MeV Gal:2013dca ; Gal:2014zia , and a $D_{12}\pi$ molecular-like model, where the mass and width of the resonance are pre-fixed to be $2370~{}\rm{MeV}$ and $70~{}\rm{MeV}$, respectively  Platonova:2014rza ; Platonova:2012am . Although some of the experimental data, like its mass and double pion decays, can be explained by using either scheme, the described structures of $d^{*}$ are quite different. Therefore, it is necessary to seek some other physical observables which would have distinct values for the different interpretations so that with the corresponding measured experimental data one would be able to justify which one is more reasonable.

It is known that the electromagnetic form factors are the indispensable physical quantities to show the internal structure of a complicated system. The electromagnetic form factors of a nucleon, for example, provide the charge and magnetic distributions inside the nucleon. The accurately measured charge radius of the proton may justify the structure of the nucleon. Consequently, the electromagnetic form factors of a the higher spin particle are also a discriminating quantity for different approaches. In particular, for the $d^{*}$ state, if there is a considerably large hidden-color component (HCC) in it, we have found that, although such a component does not contribute to its hadronic strong decay in the leading-order calculation, it plays a rather important role in the charge distribution calculation Dong ; Dong1 ; Dong2 ; Dong:2017mio , and the obtained charge distribution with a compact 6-quark scenario is quite different from that having a $D_{12}\pi$ (or $\Delta\pi N$) structure  Dong:2017mio ; Dong:2018emq . Other physical quantities, like the $d^{*}$ production in the $e^{+}e^{-}$ annihilation, and constituent quark counting rule in the high energies may also provide some other information for its structure Lu:2018gtk ; Dong:2019stt .

Another physical quantity to explore the structures of the nucleon excitations, is the electromagnetic transition amplitudes in the $\gamma$-nucleon ($\gamma$-N) interaction, such as the $\gamma N\to N^{*}$ process. There are many calculations for the electromagnetic transition amplitudes of the nucleon excitations in the $\gamma N$ process, for example in $\gamma~{}N\to\Delta(1232)$, $\gamma N\to S_{11}(1535)$, and $\gamma N\to D_{13}(1520)$ et al.  Close:1989aj ; Li:1990qu ; Giannini:1990pc ; Capstick:1992xn ; Capstick:1992uc ; Pascalutsa:2006up ; Santopinto:2012nq ; Warns:1989xr ; Warns:1989ie ; Bijker:1994yr ; Ferraris:1995ui ; Dong:2001js . Those amplitudes can also tell the nature of those nucleon resonances. However, to extract them, one has to measure the physical process, such as pion photo-production, or meson photo-production  Chiang:2001as ; Kamalov:2001qg ; Yang:1985yr ; Yang:1989si ; Nozawa:1989gy . It is also possible to get the information of the transition amplitudes from the photo-absorption cross section on the nucleon target  Stoler:1993yk ; Stuart:1996zs ; Dong:1997pv ; Drechsel:2004ki ; Aznauryan:2009da . There are some experimental data on the cross sections of the photo-absorption for the ${}^{1}H$, ${}^{2}H$, and ${}^{3}He$  MacCormick:1996jz ; MacCormick:1997ek targets and also for some nuclei of $Li$, $Be$, $C$, $Al$, $Pb$, $Sn$, and $U$ Bianchi:1994ax ; Bianchi:1995vb ; Arndt:2005wk , which can tell the nuclear medium effects.

Analogy to the study of the transverse helicity amplitudes, like $A_{1/2}$ and $A_{3/2}$ of $\Delta$ resonance in $\gamma+N\to\Delta$, here, we show a model-dependent calculation for the real photo-absorption cross section on the deuteron at the $d^{*}$ energy region. The contribution by the $d^{*}$ resonance for $\gamma+d\to d^{*}$, which is directly associated with the matrix elements of the electromagnetic interaction, is explicitly shown. It should be stressed that in the single baryon (three-quark system) case, the transition amplitudes are contributed by tree (or leading) diagrams due to the photon-quark coupling. But, in the $d^{*}$ case, the amplitudes are obtained from the contributions of sub-leading diagrams, which is very similar to those in the case of the single-pion decay of $d^{*}$ Dong2017 due to the fact that the deuteron is mainly composed of a proton and a neutron, whereas the $d^{*}(2380)$ resonance is made up of about $31\%$ $\Delta\Delta$ configuration and about $69\%$ $C_{8}C_{8}$ hidden-color configuration in our scenario. It should be mentioned that a very recent analysis Bashkanov:2019mbz shows that the experiment of intense photoinduced reaction on deuteron can provide abundant information for the structural characteristics, such as the size, the magnetic dipole and quadrupole moments, as well the deformation, of $d^{*}$.

This paper is organized as follows. In Sect. II, the hypothetic structure and the corresponding wave function of the $d^{*}$ resonance in the extended chiral SU(3) constituent quark model is briefly introduced. Sect. III is devoted to the calculation of the matrix elements of the electromagnetic interaction for the process $\gamma+d\to d^{*}$. The relevant photo-absorption cross section on deuteron contributed by $d^{*}$ is given in Sect. IV. Finally a short summary is presented in Sect. V.

II Structure and wave function of $d^{*}(2380)$ in the extended chiral constituent quark model

In 1999, a $\Delta\Delta+C_{8}C_{8}$ structure of the $d^{*}$ state with \big{(}I(J^{P}))=(0(3^{+})\big{)} (where $I$, $J$, $P$ are isospin, spin, and parity of the system, respectively,) was proposed in Ref. Yuan and its binding energy and root mean square radius (RMS) were predicted. Recently, a series of sophisticated calculations on the structure and decay characteristics of $d^{*}$ had further been performed, and the obtained mass, all the partial decay widths and total width of $d^{*}$ are all consistent with the observed data. Then, a picture of a compact structure, an exotic hexaquark dominated state, was deduced  Yuan ; Huang:2014kja ; Huang:2015nja ; Dong ; Dong1 ; Dong2 ; Dong:2017mio . In order to make this conclusion more meaningful, a so-called extended chiral SU(3) constituent quark model (ECCQM) that provides the basic effective quark-quark interactions caused by the exchanges of the chiral fields, including pseudo-scalar, scalar and vector mesons, and one gluon, as well as by the quark confinement, was employed in the dynamical calculations and in the quark degrees of freedom. In terms of this ECCQM, in which the model parameters are determined by the stability conditions and the masses of the ground state baryons, the static properties of baryons, the binding energy of deuteron, the phase shifts of the $N$-$N$ scattering and the cross sections of the N-hyperon (N-Y) interactions can be well reproduced showing the predictive power of ECCQM Yu:1995ag ; Zhang:1997ny .

Specifically, the structural calculation was carried out in the well-established Resonating Group Method (RGM), which has frequently been applied to the studies of nuclear physics and hadronic physics, especially where the clustering phenomenon does exist Tang:1977tw ; Oka:1981rj ; Faessler:1983yd ; Lacombe:2002di ; Faessler111320 ; Faessler111321 ; Faessler111322 ; Oka1984 ; Shimizu:1989ye ; Yamauchi:1992fh . In our compact structure, the trial wave function of this six-quark system with two-configuration, $\Delta\Delta$ and $C_{8}C_{8}$, can be written as

[TABLE]

where $S$, $T$ and $C$ represent the quantum numbers of the spin, isospin and color, ${{\cal A}=1-{\sum\limits_{i(\in A),~{}j(\in B)}}P_{ij}^{OSFC}}$ is an anti-symmetrization operator with $P_{ij}^{OSFC}$ denoting the exchange operator which exchanges the $i$-th quark belonging to the cluster A and $j$-th quark pertaining to the cluster B in the orbital, spin, flavor and color spaces, $\hat{\phi}^{A(B)}_{\Delta{\rm(or\,C_{8})}}$ depicts the anti-symmetrized internal wave function of the three quark cluster A(B) for either $\Delta$ or $C_{8}$ with ${\vec{\xi}}_{i}$ ($i=1,2~{}(4,5)$) being its internal Jacobi coordinates, $\mu_{\Delta{\rm(or\,C_{8})}}^{A(\rm B)}$ stands for an aggregate of the quantum numbers of the spin, isospin and color of the cluster A(B) for either $\Delta$ or $\rm C_{8}$ with $[{\it S,I,C}]_{\Delta(\rm C_{8})}=[3/2,3/2,(00)~{}(\,(3/2,1/2,(11)\,)\,)\,]$ for the $\Delta(\rm C_{8})$ cluster, and $\eta_{\Delta\Delta{\rm(C_{8}C_{8})}}$ is the relative wave function between the A and B clusters. $\eta_{\Delta\Delta{\rm(C_{8}C_{8})}}$ can be determined by dynamically solving the RGM equation Huang:2014kja ; Huang:2015nja . The reason for including a $C_{8}C_{8}$ hidden-color configuration is that as energy increases, the two $\Delta$ clusters can get closer, and they may be excited into two-colored clusters, and the two colored cluster can form a color singlet state. Consequently, such an additional configuration is QCD-allowed in enlarging the Fock space for a better description of the two-baryon system. In fact, this kind hidden-color configuration has frequently been employed to study structures of exotic hadrons, such as tetraquarks  Brodsky:2014xia ; Brodsky:2015wza ; Lebed:2018jcr ; Brambilla:2014jmp ; Maiani:2004vq ; Maiani:2014aja ; Karliner:2006hf .

However, the resultant wave functions of the two configurations shown in Eq. (II) are not orthogonal to each other. Physically, it means that the obtained $\Delta\Delta$ wave function does not only include the contributions from the non-hidden-color component, but also contains those from the hidden-color component due to the antisymmetrization operation. To orthogonalize the wave functions of these two configurations and make the numerical calculations much simplified and feasible without missing most of the important effect of anti-symmetrization, we further simplify the RGM wave function of the system by using the channel wave function with the projection procedure. This method is often used in nuclear physics and hadronization in hadron physics  Kusainov91 ; Glozman93 ; Stancu97 . Finally, we modify Eq. (II) as an effective wave function as

[TABLE]

with $ch=\Delta\Delta$ and $C_{8}C_{8}$ denoting the constituents of the configuration, $m_{d^{*}}$ representing the magnetic quantum number of spin $S_{d^{*}}$, $pw=l=0$ and $2$ depicting the $S$ and $D$ partial waves ($pw$) between the two clusters, respectively, and $m_{l}$ being its magnetic quantum number. Now, these four channel wave functions are orthogonal to each other. There are two points should be mentioned: (1) This treatment is just an approximation. The inaccuracy of such effective wave function is expected to be about 20% compared to the wave function obtained in the rigorous RGM dynamical calculation. This is because that due to the anti-symmetrization procedure, more configurations other than the initially selected $\Delta\Delta$ and $C_{8}C_{8}$ that span our model space are generated. Considering the uncertainty in the constituent quark model caused by non-perturbative QCD (NPQCD), we believe that the contribution from those spare configurations is not so important. (2) The $D$-wave is ignored in the calculations for the strong decay and charge distribution, because it is negligibly small. However, it may contribute to the higher multi-pole form factors, such as $E2$, and $M3$ since those values are closely related to the matrix elements of the high-rank operators Dong:2018emq . The more detailed information about the wave function of $d^{*}$ is referred to Ref. Dong:2017mio ; Dong:2018ryf .

III Electromagnetic Transition Form Factors of $\gamma+d\to d^{*}(2380)$

When one deals with the transverse transition amplitudes of the nucleon excitations, like $\Delta(1232)$, $P_{11}(1440)$, $S_{11}(1535)$, and $D_{13}(1520)$, in the process of $\gamma N\to N^{*}$, the spin projection of the initial nucleon (spin-1/2 particle) can be antiparallel and parallel to the spin projection of the incoming photon ($\epsilon_{\gamma}=+1$), and therefore we have two transition amplitudes $A_{1/2}$ and $A_{3/2}$ (or two helicity amplitudes in the real photon case $Q^{2}=0$), respectively. Those amplitudes are the matrix elements of the electromagnetic interaction. When we discuss the electro-production amplitudes, the virtual photon is considered and the longitudinal transition amplitude $S_{1/2}$ should be included as an additional transition amplitude, except for the two transverse ones. In the three constituent quark model for baryons, these three amplitudes are usually calculated in the Breit-frame, and the momentum and energy of the incoming photon are defined as Capstick:1992xn ; Capstick:1992uc

[TABLE]

where, $M_{N}$ and $M_{X}$ are the masses of the nucleon and nucleon excitation, and $Q^{2}=-(k_{0}^{2}-\vec{k}^{~{}2})$ depicts the squared momentum transfer.

Analogy to the calculation of transition amplitudes of $\gamma N\to N^{*}$, we perform a calculation for the transition of $\gamma d\to d^{*}$. Here we employ the Breit frame as well, where the incoming photon has a four-momentum $k=(k_{0},\vec{k})$, then the initial deuteron and final $d^{*}(2380)$ have three-momentum of $-\vec{k}/2$ and $\vec{k}/2$, respectively. We also assume that the deuteron can be reasonably regarded as a weakly bound state of a proton and a neutron, and our $d^{*}$ is mainly composed of two components, one is a hidden-color cluster $C_{8}C_{8}$ (with a large fraction of about $69\%$) and a $\Delta\Delta$ cluster with a relatively small fraction of about $31\%$. Therefore, in the calculation of the $\gamma d\to d^{*}$ transition, or of the matrix element of the electromagnetic interaction between the initial deuteron and final $d^{*}$, photo-deuteron in the leading-order approximation cannot directly reach to the final $d^{*}$ state, and its contribution vanishes. Then, we have to consider next to leading-order (NLO) terms, where exist the intermediate nucleon state and the pion exchange between the two clusters. This feature is similar to our calculation of the single pion decay partial width of $d^{*}$ Dong2017 . The possible diagrams of the NLO contribution where the photon couples directly to the upper cluster are drown in Fig. 1 and the momenta are those in the known Breit frame. Of course, the similar diagrams where the photon couples directly to the lower cluster also contribute, and these contributions are included in the calculation as well. Figs. 1 (a,b,c) and Figs. 1 (d,e,f) exhibit that the incoming photon acts on the upper cluster before and after one-pion-exchange occurs, respectively.

In the case of $\gamma+d\to d^{*}$ where the deuteron is the target, since the initial deuteron is a massive spin-1 particle with three spin projections, we have three independent transition amplitudes, two transverse amplitudes with the deuteron polarizations antiparallel and parallel to the incoming real photon, and one amplitude with the longitudinal polarization of the initial deuteron. Then, the transition amplitudes of the deuteron contributed by $d^{*}$ are ${\cal A}(m_{d})$ (with $m_{d}=\pm 1,0$ being the spin polarizations of the deuteron). So, in the real photon limit, in terms of our model wave function, we can investigate the helicity amplitudes $A_{m_{d}}$, and consequently the total photoabsorption cross section contributed by $d^{*}$.

Of course, the $\gamma+d\to d^{*}$ process can also be studied in terms of the multipoles. For the $\gamma N\to\Delta$ transition, with $J_{i}=1/2$ and $J_{f}=3/2$ being the angular moments of the initial state N and the final state $\Delta$, we have $M1$ and $E2$ multipole transitions, since the relation of $\mid J_{f}-J_{i}\mid\leq L\leq J_{f}+J_{i}$ gives $L=1$ and $2$ with $L$ denoting the order of the multipole transition and parities of the initial nucleon and final $\Delta$ being both positive (see Refs.  Capstick:1994ne ; Arenhovel:1990yg ; Pascalutsa:2006up for details). For the process of $\gamma+d(1^{+})\to d^{*}(3^{+})$ concerned, if we assume $d$ and $d^{*}$ as point particles, the above relation gives $L=2$, $3$ and $4$, which link to $E2$, $M3$ and $E4$ multipole transitions, respectively. One can obtain the connections between the helicity amplitudes and multipole transitions from simplified physical arguments. The $E2$ ($2^{+}$) transition happens when the projections of photon’s spin ($J_{\gamma}=1$) and angular momentum ($l_{\gamma}=1$ with $l_{\gamma}=L-1$ in the electric $L$-pole (EL) transition) are aligned to that of the deuteron total spin ($m_{d}=+1$), and the $E4$ ($4^{+}$) transition appears when both the projections of photon’s spin ($J_{\gamma}=1$) and angular momenta ($l_{\gamma}=3$) are antiparallel to that of deuteron total spin ($m_{d}=-1$). In addition, the $M_{3}$ transition is clearly associated to the $m_{d}=0$ state. Details for the transition Lagrangian of $\gamma+d\to d^{*}$ can be referred to Ref. Scadron:1968zz .

Now, let’s return to the framework of the helicity amplitudes. In the non-relativistic approximation, the matrix elements for Fig. 1 (a,c,f) can be written as

[TABLE]

where the superscript $"Int."$ specifies the considered intermediate state $N$ or $\Delta$, $\Psi_{d(d^{*})}$ is the wave function of the deuteron ($d^{*}$), \Big{[}T^{Int.}\Big{]}_{(a,c,f)} represents the isospin factor of Figs. 1(a,c,f), \Big{[}S^{Int.}(m_{d})\Big{]}_{(a,c,f)} denotes the spin factor with the projection $m_{d}$ of the spin of deuteron (the polarization of real photon has been chosen as $\epsilon_{\gamma}=+1$), and \big{[}{\mathcal{M}}^{\,\,\gamma N\to Int.}\big{]} stands for the convention electromagnetic transition amplitude of a three-quark nucleon in the process of $\gamma N\to N^{*}$, which has already been calculated in the constituent quark model, and its explicit forms can be found in Refs. Capstick:1992xn ; Capstick:1992uc ; Close . The energy denominators in eq. (4) can be expressed as

[TABLE]

[TABLE]

and

[TABLE]

respectively, and the product of the strong transition coupling of $\pi B\to B^{\prime}$ in eq. (4) for the corresponding sub-diagrams can be denoted by

[TABLE]

In Eqs. (5-7), the superscripts ”(1)” and ”(2)” stand for the upper and lower $\Delta$ clusters. In the constituent quark model, those couplings in eq. (8) have been explicitly discussed in Refs. Close ; Riska:2000gd .

Similarly, the matrix elements for the latter three subdiagrams in Figs. 1(b), 1(d), and 1(e) in Fig. 1 are written as

[TABLE]

with

[TABLE]

[TABLE]

and

[TABLE]

and

[TABLE]

Finally, the total electromagnetic transition amplitude is summarized as

[TABLE]

The relevant isospin and spin factors of \big{[}T^{Int.}\big{]}_{a,b,c,d,e,f} and \big{[}S^{Int.}(m_{d})\big{]}_{a,b,c,d,e,f} in eqs. (4) and (9) are given in Tables 1 and 2. It should be mentioned that the spin factor here includes all the necessary high-partial wave contributions due to the intermediate pion exchange in Fig. 1. For instance, in a case with the simplest $E2$ transition and the $S$-wave $pn$ configuration of $d$, when one of the nucleon in $d$ is excited by a photon to $N^{*}$ leaving another nucleon as a ”spectator”, the only allowed nucleon resonance is $N^{*}(5/2^{+})$. However, due to the pion-exchange in the time-ordered diagram, the nucleon can be excited to $N^{*}(1/2^{+})$ in $D$-wave to another nucleon, leaving the total spin of $d^{*}$ being $3$. Namely, in our numerical calculation, the small $D$-wave component in the relative wave functions of the deuteron ($5-6\%$) and $d^{*}$ (about $0.5-0.6\%$ in the $\Delta\Delta$ configuration and about $0.0-0.02\%$ in the $C_{8}C_{8}$ configuration) are explicitly taken into account (see for example Ref. Dong:2018emq for the $D$-wave of $d^{*}$).

IV Photo-absorption cross sections on deuteron in $d^{*}$ energy region

It is well known that the photon-absorption cross sections on a nucleon can provide transparent information for the nucleon resonances, such as their transition amplitudes. There have been many analyses of the real photon-absorption cross sections in the energy region from the pion photo-production point to $2~{}\rm{GeV}$ Stoler:1993yk ; Stuart:1996zs ; Dong:1997pv ; Drechsel:2004ki ; Aznauryan:2009da . Unlike the case of the three-quark nucleon, there are three spin-dependent photo-absorption cross sections on the polarized deuteron with the polarization of $\sigma_{m_{d}}$ (with $m_{d}=\mp 1,0$), namely, they are the absorption cross sections corresponding to the polarized photon with the helicity being antiparallel and parallel to the transversely polarized deuteron and that to the longitudinally polarized deuteron, respectively. Therefore, the photon-absorption cross sections ($Q^{2}=0$) from the contribution of a resonance can be calculated by Stoler:1993yk

[TABLE]

where $\sum_{R}$ means the summation over all the resonances in the energy region, $W$ is the center-of-mass energy, and $\Gamma$ is the total decay width of the resonance. In the low-$Q^{2}$ range, it is adequate to represent the resonance shape by a simple non-relativistic Breit-Wigner form of $R(W,\Gamma)$

[TABLE]

where $M_{d}$, and $M_{R}$ stand for the masses of deuteron and $d^{*}(2380)$, respectively. The total photo-absorption cross section on deuteron from the contribution of the $d^{*}(2380)$ resonance can be written as

[TABLE]

At the resonant point of $W=W_{R}$, it can be further expressed as

[TABLE]

Based on the electromagnetic matrix elements obtained in Sect. III, the photo-absorption cross sections on deuteron provided by the contribution of the $d^{*}$ resonance can be calculated, and corresponding results are plotted in Fig. 2.

In this figure, we find that the contribution from $d^{*}$ to the real photon-absorption cross sections is in the order of ten nano-barn (nb) at the resonant point of $d^{*}$. The total estimated cross section is about $\sigma^{d^{*}}(M_{d^{*}};T)\sim 9.35~{}\rm{nb}$, and the $\sigma^{d^{*}}(M_{d^{*}};m_{d})$ are $9.92~{}\rm{nb}$, $11.56~{}\rm{nb}$, and $6.49~{}\rm{nb}$ for $m_{d}=+1,0,-1$, respectively. The corresponding helicity amplitudes of the transition are $\mid{\cal A}_{1}\mid=0.754\times 10^{-3}~{}\rm{GeV}^{-1/2}$, $\mid{\cal A}_{0}\mid=0.812\times 10^{-3}~{}\rm{GeV}^{-1/2}$, and $\mid{\cal A}_{-1}\mid=0.608\times 10^{-3}~{}\rm{GeV}^{-1/2}$, respectively. Those transition amplitudes are much smaller than the helicity amplitudes of $\Delta(1232)$ resonance ($A_{1/2}^{\Delta}=(-141\pm 5)\times 10^{-3}~{}\rm{GeV}^{-1/2}$, $A_{3/2}^{\Delta}=(-258\pm 19)\times 10^{-3}~{}\rm{GeV}^{-1/2}$). Moreover, the photon-absorption cross section on deuteron from the contribution of $d^{*}$ in our scenario is almost $5-6$ orders of magnitude smaller than that of about $500~{}\rm{\mu b}$ on nucleon from the contribution of $\Delta$ resonance (see for example Ref. Drechsel:2004ki ). The reasons for the remarkable suppression are the following: 1) The lowest non-vanishing diagrams shown in Fig. 1 are in the next-leading order, since the incoming photon excites one of the nucleon to the $\Delta$ resonance and another $\Delta$ should emerge by exciting another nucleon via the one pion exchange, which is similar to the diagrams given in Kanda:2015 . 2) The wave function of $d^{*}$ in our suggested structural model contains two components, $\Delta\Delta$ and $C_{8}C_{8}$, and the probability of the former is only about $1/3$ of the total. 3) Even in the $\Delta\Delta$ case, the effect that one of the $\Delta$s is bound by another $\Delta$ makes the amplitude suppressed. 4) The constituent quark model is not good enough to explain the extracted transition amplitudes $A_{1/2,3/2}$, even for the $\Delta$ resonance, from the experimental measurements. It can only reproduce about $2/3$ of the data. One expects that the pion meson cloud might play an important role in understanding the data Dong:2001js ; Kaelbermann:1983zb ; Bermuth:1988ms ; Lu:1997sd ; Lu:1996rj ; Dong:1999cz . In addition, our obtained ${\cal A}_{\pm,0}$ are complex numbers, unlike the real number in the tree diagram calculation for a single nucleon excitation.

It should be mentioned that several experiments on the $\gamma d$ at the $d^{*}$ energy region have been carried out at MAINZ Gunther:2017ngt ; Bashkanov:2018ftd . Moreover, an experiment on the $\gamma d\rightarrow\pi^{0}\pi^{0}d$ at the incident energy of $0.55\sim 1.15~{}\rm{GeV}$ has been carried out in the Research Center for Electron Photon Sciences (ELPH) at Tohoku University, Japan, and the new data on dibaryon were released. It is shown that a signal of the $d^{*}(2380)$ state with the width of $70~{}\rm{MeV}$ was clearly exhibited in the mass spectrum of $\pi^{0}\pi^{0}d$, and the corresponding cross section at this resonant point is about $28~{}\rm{nb}$  Ishikawa:2016yiq ; Ishikawa:2018 . Compared with the observations, our estimated cross section for the total photon absorption is about $20$ times smaller than the data, since the branching ratio of $d^{*}\to\pi^{0}+\pi^{0}+d$ is about $~{}14\%$. It should be stressed that in this calculation, we only consider the direct coupling of a photon to one of nucleons inside the deuteron. It seems insufficient to describe such process. This defect could be partially attributed to the following reasons. In our calculation of deuteron, the $C_{8}C_{8}$ and $\Delta\Delta$ components are very small. Although a colorless-nucleon pair cannot be directly converted to a colored-$C_{8}$ pair by a photon in the leading order approximation, a very small hidden-color component in deuteron can change to a dominant hidden-color component in $d^{*}$ by the action of a photon on the deuteron. The large change in the fraction of the hidden-color component when $d$ is converted to $d^{*}$ implies that lack of this mechanism would partially affect the loss of the photo-absorption cross section. Another reason could be that due to the existence of the external electromagnetic field, one needs the coupling of the photon with the meson exchange current during the excitation of $d^{*}$ by the photon in the $pn\to\Delta\Delta$ transition Meyer:2001js ; Buchmann:1997em ; Meyer:1998td ; Yamauchi:1991hu ; Kotlyar:1987hy . This kind of interaction, might provide a sizeable contribution to the $\gamma d\to d^{*}$ transition. Moreover, the action of a photon on the spin-1 $\Delta\Delta$ component in deuteron can also generate the spin-3 $\Delta\Delta$ component in $d^{*}$. Although the $\Delta\Delta$ component in $d$ is very small, this mechanism might also provide a measurable effect. In addition, due to the pion-exchange in the time-ordered diagram, the nucleon can be excited to $N^{*}(1/2^{+})$ in $D$-wave to another nucleon, which might be another factor for the suppression of the photoabsorption cross section compared with the data. Of course, there might also some other factors that would affect the photoabsorption cross section, for instance other intermediate processes, the higher partial wave component in deuteron and $\Delta$, the defect of the quark model in describing the photon associated process, and etc..

Finally, in order to closely relate the theoretical results with the actual experiments, and to further obtain the information about other intermediate mechanisms and asymmetric behaviors in the transition process of $\gamma d\to d^{*}\to pn$ it is necessary to study the differential cross sections with the polarized photon and polarized target. These will be done in our future study.

V Summary

In order to understand the internal structure of the $d^{*}$ resonance discovered by CELSIUS/WASA and WASA@COSY Collaborations, two major structural schemes were proposed recently. One of them, based on the quark degrees of freedom, considers that it has a compact exotic hexaquark dominated structure due to the quark exchange effect, and the other, in terms of hadronic degrees of freedom, believes it as a molecular-like hadronic state. These two structures have been tested in terms of the experimental data. Up to now, both models can explain the mass, the total width, and the partial decay widths for all the observed double pion decays of the $d^{*}$ resonance. However, for a single pion decay process, although the observed upper limit of the branching ratio can be explained by both structure models, the ways of explanation have a difference. Therefore, we need to seek other physical quantities to distinguish these two different structures for $d^{*}$. Of course, the realistic structure of $d^{*}$ might be much more complicated, for instance, our compact hexaquark dominated core as an essential ingredient is mixed with other ingredients, such as a $D_{12}\pi$ cloud Gal1 . This picture just looks like the commonly believed nucleon where a three-quark core is surrounded by the meson cloud.

The aim of this paper is to find the contribution of $d^{*}$ to the total photoabsorption cross section on deuteron target by using the $d$ and $d^{*}$ wave functions obtained in our scenario. From a theoretical point of view, our compact picture may not be able to provide enough contribution. This is because that in the calculated diagrams, we only consider the $\Delta\Delta$ configuration of $d^{*}$, whose probability is about two times smaller than that of the $C_{8}C_{8}$ component, since the colorless-nucleon pair cannot be converted to the colored cluster pair (hidden-color configuration) by a photon in the lowest order approximation. Then, the estimated total photoabsorption cross section of $\gamma d\to d^{*}$ at the resonant point of $d^{*}$ is just about ten $nb$ which is much smaller compared with the data. Furthermore, we would mention that the quark model predictions for the helicity amplitudes of the single $\Delta$ resonance are about $30\%$ smaller than the experimental data (see Refs. Dong:2001js ; Kaelbermann:1983zb ; Bermuth:1988ms ; Lu:1997sd ; Lu:1996rj ; Dong:1999cz ). Even if we can make up for the underestimation of the quark model in some way, the photoabsorption cross section of $\gamma d\rightarrow\pi^{0}\pi^{0}d$ from the $\Delta\Delta$ component of $d^{*}$ is around $1\sim 2~{}nb$, which is still about one order of magnitude smaller than the currently data of about $28~{}\rm{nb}$ measured by ELPH and A2 at MAINZ Gunther:2017ngt ; Bashkanov:2018ftd ; Ishikawa:2016yiq ; Ishikawa:2018 . The small signal might be submerged in the contribution from the background, which comes from the other mechanisms, for instance, the conversion of the very small hidden-color configuration in deuteron to a dominated hidden-color configuration in $d^{*}$, the coupling of the photon to the meson exchange current, and so on. In particular, as pointed out in the recent paper Bashkanov:2019mbz , the deformation of the $\Delta$ wave function may play an important role in the $\gamma d\to d^{*}$ transition as well. Inclusion of these mechanisms would enhance the photoabsorption cross section, so that the cross section of the $\gamma d\to d^{*}$ transition may become visible. It should be mentioned that although our chiral quark model has already reasonably reproduced the deformation of wave function and the $E2/M1$ ratio for the $\Delta$ resonance Shen:1997jd , for roughly estimating this transition rate of $\gamma d\to d^{*}$, we ignore the deformation of the $\Delta$ wave function temperedly. It seems that this deformation effect in the single $\Delta$ resonance is larger than that from the $D-$ wave component between the two clusters. Therefore, those mechanisms, especially the contribution from the deformations of the single $\Delta$ clusters, should be carefully investigated in future. In addition, we should mention that by considering the real photo-production process, one might also obtain the information of the magnetic moment through the excitation of the $P$-shell nucleon in nucleon pair by a photon via a $M1$ transition as well Bashkanov:2019mbz ; Bashkanov:2018 .

Acknowledgment

This work is supported by the National Natural Sciences Foundations of China under the Grant Nos. 11475192, 11475181, 11521505, 11565007, and 11635009, the Sino-German CRC 110”Symmetries and the Emergence of Structure in QCD” project by NSFC under the grant No.11621131001, the Key Research Program of Frontier Sciences, CAS, Grant No. Y7292610K1, and the IHEP Innovation Fund under the grant No. Y4545190Y2. Authors thank the fruitful discussions with Mikhail Bashkanov, and Yubing Dong thanks Fei Huang for providing the wave functions of $d^{*}$.