Pion-less effective field theory on low-energy deuteron electrodisintegration

In view of its relation to Big Bang nucleosynthesis and a reported discrepancy between nuclear models and data taken at S-DALINAC, electro-induced deuteron breakup ${}^2\mathrm{H}(e,e^{\text{'}}p)n$ is studied at momentum transfer $q<100$ MeV and close to threshold in the low-energy nuclear effective field theory without dynamical pions, EFT($\pi /$). The result at next-to-next-to-leading order (N${}^2\mathrm{LO}$) for electric dipole currents and at next-to-leading order (NLO) for magnetic ones converges order-by-order better than quantitatively predicted and contains no free parameter. It is at this order determined by simple, well-known observables. Decomposing the triple differential cross section into the longitudinal-plus-transverse ($L+T$), transverse-transverse (TT), and longitudinal-transverse interference ($\mathit{LT}$) terms, we find excellent agreement with a potential-model calculation by Arenhövel and co-workers, based on the Bonn potential. Theory and data also agree well on ${\sigma }_{L+T}$. There is however no space on the theory side for the discrepancy of up to $30\%(3\sigma )$ between theory and experiment in ${\sigma }_{\mathit{LT}}$. From universality of EFT($\pi /$), we conclude that no theoretical approach with the correct deuteron asymptotic wave function can explain the data. Undetermined short-distance contributions that could affect ${\sigma }_{\mathit{LT}}$ enter only at high orders (i.e., at the few-percent level). We notice some issues with the kinematics and normalization of the data reported.

Figure 1

Kinematics of deuteron electrodisintegration, reproduced with the kind permission of the authors of Ref. [5]. The electron kinematics refers to the laboratory frame, while the proton variables are defined in the center-of-mass frame of the two-nucleon final state.

Figure 2

The LO electric contributions to the hadronic current. The double line denotes the spin-triplet dibaryon intermediate state. There is no NLO contribution.

Figure 3

The order-$Q^2$ (N${}^2\mathrm{LO}$) electric contributions. The SD-mixing vertex proportional to $C_{\mathit{SD}}$ is denoted by a circle; further notation as in Fig. 2.

Figure 4

The magnetic contributions to the hadronic current at LO (m1–m3) and NLO (m4). The blob denotes a magnetic photon coupling via the isovectorial magnetic moment, the square the dibaryon-photon interaction [Eq. (2.17)] proportional to $L_1$, and the dashed double line the spin-singlet dibaryon intermediate state; further notation as in Fig. 2.

Figure 5

Examples of the triple-differential cross section for $E_X^{\mathrm{lab}}=9\hspace{1em}\mathrm{MeV}$ at $E_0^{\mathrm{lab}}=85$ MeV, compared to S-DALINAC data [5] (with combined statistical and systematic error bars) and to the result by Arenhövel and co-workers [4, 5, 7] (squares). (a) Interpretation of “$E_X=9\hspace{1em}\mathrm{MeV}$” as given in the total c.m. frame, the $\mathit{pn}$ c.m. frame, or the laboratory frame. (b) EFT($\pi /$) at $E_X^{\mathrm{lab}}=9$ MeV (solid line) and $E_X^{(\mathit{pn})-\mathrm{c}.\mathrm{m}.}=9$ MeV (i.e., $E_X^{\mathrm{lab}}=10.6$ MeV; dashed). The dotted curve is the double-differential cross section per MeV, obtained by integrating over the bin $E_X^{\mathrm{lab}}\in [8;10]\hspace{1em}\mathrm{MeV}$; the difference from the solid line is almost invisible. (c) Comparison on a logarithmic scale of the EFT($\pi /$) contributions: electric transitions up to LO and N${}^2\mathrm{LO}$, respectively (recall that NLO is zero), and magnetic transitions up to NLO.

Figure 6

EFT($\pi /$) results for the differential cross section, compared to data and calculations by Arenhövel and co-workers. (a) Data set at smallest energy and momentum transfer: $E_0^{\mathrm{lab}}=50\hspace{1em}\mathrm{MeV},\hspace{1em}{E}_X^{\mathrm{lab}}=9\hspace{1em}\mathrm{MeV}$. (b) Data sets at largest energy and momentum transfer: $E_0^{\mathrm{lab}}=85\hspace{1em}\mathrm{MeV},\hspace{1em}{E}_X^{\mathrm{lab}}=15\hspace{1em}\mathrm{MeV}$. (e), (f) Data set at $E_0^{\mathrm{lab}}=85\hspace{1em}\mathrm{MeV},\hspace{1em}{E}_X^{\mathrm{lab}}=\{11;13\}\hspace{1em}\mathrm{MeV}$. Comparisons of EFT($\pi /$) contributions are in panels (c) and (d) included for the smallest and largest energy and momentum transfer: electric transitions up to LO and N${}^2\mathrm{LO}$, respectively (NLO is zero), and magnetic transitions up to NLO.

Figure 7

Decomposition of the triple-differential cross section into (top to bottom) longitudinal-plus-transverse ($L+T$), longitudinal-transverse ($\mathit{LT}$), and transverse-transverse ($\mathit{TT}$) parts at $E_0^{\mathrm{lab}}=85$ MeV, normalized to ${\sigma }_{L+T}$ at ${\Theta }_p=0$. Left column: $E_X^{\mathrm{lab}}=9$ MeV; right column: $E_X^{\mathrm{lab}}=11$ MeV. No data are available for the $\mathit{TT}$ interference cross section.

Figure 8

The same as Fig. 7, but for $E_X^{\mathrm{lab}}=13$ MeV (left column) and $E_X^{\mathrm{lab}}=15$ MeV (right column).

Figure 9

$E_X^{\mathrm{lab}}$ dependence of ${\sigma }_L+{\sigma }_T$ at ${\Theta }_p={180}^{°}$, normalized to ${\Theta }_p=0^{°}$ (a) and before normalization (b), and (c) ${\sigma }_L+{\sigma }_T$ at ${\Theta }_p=0^{°}$ (to which all data are normalized). All graphs refer to $E_0^{\mathrm{lab}}=85$ MeV, ${\Theta }_e^{\mathrm{lab}}={40}^{°}$.

Figure 10

$E_X^{\mathrm{lab}}$ dependence of ${\sigma }_{\mathit{LT}}$ at the minimum around ${\Theta }_p={35}^{°}$, normalized (a) and absolute values (b), both for $E_0^{\mathrm{lab}}=85$ MeV, ${\Theta }_e^{\mathrm{lab}}={40}^{°}$.

Figure 11

$E_X^{\mathrm{lab}}$ dependence of the absolute values of ${\sigma }_{\mathit{LT}}$ for the extreme cases $E_X^{\mathrm{lab}}=9\hspace{1em}\mathrm{MeV}$ (a) and $15\hspace{1em}\mathrm{MeV}$ (b) at $E_0^{\mathrm{lab}}=85$ MeV, ${\Theta }_e^{\mathrm{lab}}={40}^{°}$. Compare to the normalized data in Figs. 7 and 8.

Figure 12

Comparison to the data of Ref. [8] for ${\sigma }_{L+T}$ (a) and ${\sigma }_{\mathit{LT}}$ (b), both in the laboratory frame (also for the hadronic variables!) and normalized to ${\sigma }_{L+T}$ at ${\Theta }_p=0$.