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Delta Baryon Photoproduction with Twisted Photons

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Delta Baryon Photoproduction with Twisted Photons Delta baryon photoproduction with twisted photons Andrei Afanasev1 and Carl E. Carlson2 1Department of Physics, The George Washington University, Washington, DC 20052, USA 2Physics Department, William & Mary, Williamsburg, Virginia 23187, USA (Dated: May 18, 2021) A future gamma factory at CERN or accelerator-based gamma sources elsewhere can include the possibility of energetic twisted photons, which are photons with a structured wave front that can allow a pre-defined large angular momentum along the beam direction. Twisted photons are potentially a new tool in hadronic physics, and we consider here one possibility, namely the photopro- duction of ∆(1232) baryons using twisted photons. We show that particular polarization amplitudes isolate the smaller partial wave amplitudes and they are measurable without interference from the terms that are otherwise dominant. 40 I. INTRODUCTION ing S1=2 to D5=2 transitions in Ca ions with quantum number changes beyond what a plane wave photon could Twisted photons are examples of light with a struc- induce. Further, the sharing of final state angular mo- tured wave front that can produce transitions with quan- mentum between internal and overall degrees of freedom, tum number changes that are impossible with plane wave when the ion was offset from the vortex line, was also photons. One can envision a number of applications in measured and matched well with theoretical studies [7,8]. the field of hadron structure that would require sources of The ∆(1232) is a spin 3/2 baryon that can be pho- toexcited from the nucleon (Fig.1) via M1 (with related twisted photons with MeV-GeV energy scales. Such en- ∗ ∗ ergies are achievable via Compton up-conversion in high- notations M1+ and GM ) and E2 (ditto with E1+ or GE) energy electron collisions with twisted optical photons transitions. For a thorough review of electromagnetic [1,2] or in twisted-photon collisions with high-energy excitation of the ∆(1232), see [9]. In simple models, the ions, as recently suggested for CERN Gamma Factory ∆ is dominantly a spatial S-state with a spin-3/2 spin [3]. Presently, HIGS facility is making important steps wave function. This can be obtained from the nucleon toward twisted-photon generation [4], opening new op- by a simple spin flip, which an M1 transition can do, portunities for nuclear physics studies. and there is a large N to ∆ M1 amplitude. The E2 tran- sition requires two units of orbital angular momentum, This article will focus on how twisted light may con- and must involve the small D-wave spatial component of tribute to measuring small but important contributions the ∆ or of the nucleon. Accurate knowledge of the E2 to the electromagnetic production of ∆(1232) baryons size would help elucidate the ∆ composition and hadron from nucleon targets. structure generally. More specifically, twisted photons are states with total angular momentum whose projection along the direction of motion can be any integer, mγ , times ~. The Poynting vector or momentum density of these states swirls about a vortex line, and the intensity of the wavefront is typi- cally zero or very small on the vortex line. (Indeed, the \hole" in the middle of the wavefront can find applica- tions seemingly unconnected to the swirling of the states, as in stimulated emission spectroscopy studies; see [5].) In photoabsorption, the photon's projected angular arXiv:2105.07271v1 [hep-ph] 15 May 2021 momentum mγ is transferred to the target system and FIG. 1. Photoexcitation of ∆(1232) baryon on a proton tar- may be shared between internal excitation of the final get. state and orbital angular momentum of the final state's overall center-of-mass. That the final internal angular momentum projection can differ by mγ units from that Currently the particle data group [10] quotes an ∗ ∗ of the target allows transitions quite different from the GE=GM ratio in the 2-3 % range, based on plane wave plane wave case, where the photon necessarily transfers photon cross section measurements where the E2 is nei- mγ = Λ = ±1 to the final excitation. Here Λ is the he- ther the only nor the dominant contribution. We shall licity or spin along the direction of motion of the photon show that twisted photons can in principle produce sig- and that Λ = ±1 only is a standard fact for plane waves. nals there the E2 is the only contributor. That large projected angular momentum transfer In much of what follows, it is more natural to describe works in practice for atoms has been shown experimen- plane wave ∆ photoproduction using helicity amplitudes tally and reported in [6], where optical orbital angular M(pw), where Λ is the helicity of the photon and m is the miΛ i momentum was transferred to bound electrons, excit- spin of the nucleon target along the photon's momentum 2 direction. There are two independent helicity amplitudes k x p (pw) ∗ ∗ M1=2;1 / 3 (GM + GE) ; (pw) ∗ ∗ M−1=2;1 / GM − 3GE; (1) k and others can be obtained by parity transformation M(pw) = −M(pw) : (2) −mi;−Λ miΛ Propagation Direction In this notation, our goal is to use twisted photons to isolate physically achievable situations where the helicity amplitudes combine with the M1 contributions canceling and the E2 not. The calculation of the N to ∆ transition with twisted k photons is in many ways analogous to atomic calcula- z tions. However, the most straightforward atomic calcu- lations are made in a no-recoil limit, which gives accurate FIG. 2. A twisted photon state in wavenumber space or mo- mentum space. results for targets that are quite massive compared to the photon energy. For the the N to ∆ transitions we do take account of the recoil, eventually finding that the recoil corrections are small, at the level of a few %. Also, for the Helmholtz equation. The states can be most sim- the apparently simplest atomic analog, an S to D 1=2 3=2 ply written in wave number space, or momentum space, transition in a single electron atom, the M1 does not where they can be represented as a collection of plane contribute at all to leading order. However, our crucial wave photons, each with the same value of k , where z results are based on the properties of the twisted photon z is the direction of propagation of the state, each with and on the quantum numbers and rotation properties ~ of the hadronic states. A more exactly analogous atomic the same magnitude of transverse momentum jk?j = κ, analog can be found among multi-electron atoms. In par- and hence each with the same polar angle or pitch an- ticular, there is an Boron-like example where the calcu- gle θk = arctan(κ/kz), but differing azimuthal angles φk. lated M1 and E2 amplitudes are about the same size [11], The set of wave vectors thus form a right circular cone and we have elsewhere shown how twisted photons could in momentum space, Fig.2. be instrumental in measuring these amplitudes [12]. The state is [1, 15, 16] In the following, Sec.II will contain some background dφk ~ ~ material on twisted photons, which may be skipped or ~ mγ imγ φk−ik·b ~ jκmγ kzΛbi = A0 (−i) e jk; Λi ; (3) skimmed by readers already expert. Sec. III will, for the ˆ 2π sake of beginning simply, find results for twisted photon induced N to ∆ transitions in the no-recoil limit. Sec.IV where mγ is the total angular momentum in the z- will include proper relativistic consideration of the ∆ re- direction, j~k; Λi is a plane wave photon state with coil, and will show that the corrections are visible on some plots but that they are not large. Some final com- ~ 0 0 ~ 3 3 ~ 0 ~ hk ; Λ jk; Λi = (2π) 2!δΛ0Λδ (k − k) ; (4) ments will appear in Sec.VI. Λ is the helicity of each component state, ! is the an- gular frequency of the monochromatic state, and A0 II. TWISTED PHOTONS is a normalization chosen, for example, in [1, 15] as p A0 = κ/(2π). The state written has a vortex line pass- A twisted photon is a state whose wavefront travels in ~ ing through the point b = (bx; by; 0), where we might a definite direction and which has arbitrary integer an- instead give a magnitude b and azimuthal angle φb. gular momentum along its direction of motion. Reviews With this state normalization, the electromagnetic po- may be found, for example, in [13, 14]. In addition, the tential for a plane wave photon is states studied are usually monochromatic, and at a theo- retical level, one can choose between Laguerre-Gaussian ~ ~ −ikx or Bessel versions of these states. We here use the lat- h0jAµ(x)jk; Λi = µ(k; Λ)e ; (5) ter; experience has shown that results are numerically ~ nearly the same either way, whereas analytic expressions where µ(k; Λ) is a unit polarization vector for a photon are simpler for Bessel modes. of the stated momentum and helicity. The vector poten- Bessel photons, in addition to the twistedness and tial for the twisted state in coordinate space can now be monochromaticity, are nondiffracting exact solutions to worked out, and in cylindrical coordinates and for ~b = 0 3 the components are The Hamiltonian is rotation invariant. Rotations of the nucleon states are given in terms of the Wigner func- A 0 i(kz z−!t+mγ φ) tions, Aρ = ip e 2 y imiφk X 1=2 0 R (φk; θk; 0) jN(mi)i = e d 0 (θk) jN(mi)i ; 2 θk 2 θk mi;mi × cos Jmγ −Λ(κρ) + sin Jmγ +Λ(κρ) ; m0 2 2 i (10) A 0 i(kz z−!t+mγ φ) Aφ = −Λ p e and rotations of the ∆ can be given analogously|if the 2 recoil velocity of the ∆ is neglected.
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