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 photoproduction 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 transition 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 applications 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 √ (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 momentum 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 |k⊥| = 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 ~ |κmγ kzΛbi = A0 (−i) e |k, 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, |~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 |k, Λi = (2π) 2ωδΛ0Λδ (k − k) , (4) ments will appear in Sec.VI. Λ is the helicity of each component state, ω is the angular 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 theoretical level, one can choose between Laguerre-Gaussian ~ ~ −ikx or Bessel versions of these states. We here use the lat- h0|Aµ(x)|k, 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−ωt+mγ φ) tions, Aρ = i√ e 2 † imiφk X 1/2 0 R (φk, θk, 0) |N(mi)i = e d 0 (θk) |N(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−ωt+mγ φ) Aφ = −Λ √ e and rotations of the ∆ can be given analogously—if the 2 recoil velocity of the ∆ is neglected. One obtains θ θ 2 k 2 k m −m i(m +m −m )φ × cos Jm −Λ(κρ) − sin Jm +Λ(κρ) , M = A (−i) f i e γ i f b J (κb) 2 γ 2 γ 0 mf −mi−mγ

A0 X 3/2 1/2 (pw) i(kz z−ωt+mγ φ) × d 0 (θk) d 0 (θk) M 0 . (11) Az = Λ √ e sin θk Jmγ (κρ), (6) mf ,mi+Λ mi,mi mi,Λ 0 2 mi with E~ = iωA~ and B~ = −iΛE~ . The Jν are Bessel func- The plane wave amplitude is defined from tions. For general impact parameter, let ~ρ be the trans- 0 0 (pw) ~ h∆(mf ) |H(0)| N(mi); γ(kz,ˆ Λ)i = M 0 δm0 ,m0 +Λ. verse coordinates and let ρ → |~ρ − b|. mi,Λ f i One sometimes makes a paraxial approximation, (12) wherein the pitch angle θk is small, and one keeps only The δ-function follows because all the spins and momenta the leading nontrivial terms in θk. We will usually not do in the plane wave amplitude are along the z-direction. this. Additionally, one can find exact twisted wave solu- In terms of the Jones-Scadron form factors [9, 20], tions to the Helmholtz equation where the smaller terms r (pw) 3eEγ 2 (for small θk) in Aρ and Aφ above are absent [17], with M = − (G∗ + G∗ ) , 1/2,1 2 3 M E the expenses of changing the longitudinal component and √ of not having all the component photons in momentum 3 eE r2 M(pw) = − γ (G∗ − 3G∗ ) , (13) space have the same helicity. We will find the uniformity −1/2,1 2 3 M E in helicity useful, and so stay with the forms above. p The time averaged Poynting vector is where an isospin Clebsch-Gordan factor 2/3 for the p → ∆+ transition is included, and hS i = 0, ρ 2 2 2 2 m∆ − mN 2π ω A0 Eγ = = (14) hS i = sin θ J (κρ) 2mN λ φ 2 k mγ θ θ is the photon energy to excite a ∆ from a nucleon at × cos2 k J (κρ) + sin2 k J (κρ) , (7) 2 mγ −Λ 2 mγ +Λ rest [21]. 2 2 ω A0 4 θk 2 4 θk 2 hSzi = cos J (κρ) − sin J (κρ) . 2 2 mγ −Λ 2 mγ +Λ III. TWISTED AMPLITUDES FOR ∆ PHOTOEXCITATION To close this section and prepare for the next, we work out the photoabsorption amplitude involving the twisted When a stationary proton absorbs a photon of the cor- photon in the limit where the ∆ is treated as very heavy rect energy to produce a ∆(1232), the ∆ recoils with the and its recoil velocity is neglected. The necessary ma- momentum acquired from the photon, at a not so negli- nipulations mirror the atomic physics case worked out gible speed of about 0.27c. None-the-less, we will start in [18]. We wish to obtain the amplitude by calculating the twisted photon photoexcitation amplitudes neglecting the recoil. We will find certain aspects of M = h∆(m ) |H(0)| N(m ); γ(κm k Λ~b)i , (8) f i γ z the no-recoil results are can be understood qualitatively, where H is the interaction Hamiltonian density. The nu- and others aspects can in simple ways be worked out an- cleon is at rest, and its spin projection along the z-axis alytically. In particular, that only the E2 amplitude can contribute to transitions with ∆m = mf − mi = ±2 is is labeled as mi. Similarly, we label the spin projection easy to understand. Then in the next section, we will of the ∆ along the same axis as mf . The twisted photon state can be expanded in plane waves, as in Eq. (3), show that for crucial observables, the recoil corrections and the plane wave photon states can be obtained by change the result in fractional terms by only about the rotations of states with momenta in the z-direction, square of half the recoil speed (in units of c). The no-recoil result for the amplitude is like the result ~ |k, Λi = R(φk, θk, 0) |kz,ˆ Λi , (9) in an atomic transition. It is given in terms of two independent plane wave amplitudes, a pair of Wigner func- which follows [1, 15] in using the Wick phase conventions tions, and a Bessel function, and is shown as Eq. (11) in of 1962 [19]. the previous section. 4

(a) We have prepared plots for a variety of mγ , Λ, and mγ=2,Λ=1,m Δ=1/2,m p=-1/2 mf = m∆, all for mi = mp = −1/2 and placed the results in a 4 × 6 grid in the Appendix. (Plots for mi = 1/2 are 400 Total identical if one makes the appropriate parity changes on M1 only

the other quantum numbers.) | 300 E2 only

To understand the plots, we focus on two of them, 200 shown in Fig.3(a) and (b). The ordinate for both is the Amplitude | magnitude of the amplitude, the abscissa is the displace- 100 ment of the vortex line of the twisted beam from the target proton, sweeping from the left along (say) the x- 0 axis to the right in units of photon wavelength, λ. Both -6 -4 -2 0 2 4 6 b/λ of plots are for mγ = 2, Λ = 1, and mi = −1/2. Fig.3(a) has mf = 1/2 and Fig.3(b) has mf = 3/2. The plots (b) m =2,Λ=1,m =3/2,m =-1/2 are for pitch angle θk = 0.2. γ Δ p 15 General observations valid for all these plots, and not Total speciﬁc to the N → ∆, include M1 only | 10 E2 only • on-axis observation: If the target sits on the vortex line, b = 0, the angular momentum of the photon must

Amplitude 5 all go into the internal excitation of the final state, i.e., | ∆m = mf − mi = mγ . If this cannot happen, the amplitude must be zero in the center, as in Fig.3(a) but not 0 Fig.3(b). -6 -4 -2 0 2 4 6 • off-axis observation 1: off-axis, the photon’s total an- b/λ gular momentum can be shared by the internal excitation (c) of the final state and by angular momentum of the final mγ=2,Λ=1,m Δ=3/2,m p=-1/2 state overall center of mass [22], and the amplitudes in 15 Total general are not zero for b 6= 0. M1 only

| E2 only • off-axis observation 2: When the target is away from 10 the vortex line, the target sees only a piece of the swirl, which can look more unidirectional, like a plane wave. Amplitude The selection rules reflect this and transitions possible | 5 with plane wave quantum numbers tend to have larger peak amplitudes than other transitions. This can be seen 0 by comparing the vertical scales in Figs.3(a) and (b). -6 -4 -2 0 2 4 6 b/λ Specific to the N → ∆ transition are the separate contributions of the M1 and E2 amplitudes, which are FIG. 3. Selected amplitude magnitude plots for twisted pho- shown in the plots. Most interesting is Fig3(b), where ton + proton → ∆ vs. displacement of the proton from the the M1 does not contribute at all. A measurement with photon’s vortex line, along (say) the positive and negative the final ∆ constrained to be in the projection mf = 3/2 x-axis, in units of the photon wavelength (3.65 fm). The am- state would give a direct measurement of the E2 ampli- plitude units are arbitrary but are the same for each figure. tude. Each of these plots has mγ = 2, Λ = 1, and mi = mp = −1/2. Part (a) has mf = m∆ = 1/2 and parts (b) and (c) have ∗ ∗ The plots are for GE/GM = 3%, and Fig.3(a) shows mf = m∆ = 3/2. (a) and (b) omit recoil corrections, as in the a more usual case where the M1 gives the major share Sec.III. (c) includes recoil corrections, as in Sec.IV. The E2 of the amplitude. only curve overlays or nearly overlays the Total in parts (b) and (c). The recoil corrections are visible but small. Further regarding the absence of M1 contribution to the ∆m = mf − mi = 2 transition, there are only two terms in the sum for the amplitude, Eq. (11), so it is easy to insert explicit Wigner functions and plane wave ampli- IV. RECOIL MOMENTUM CORRECTIONS tudes in terms of M1 and E2, Eq. (13), and demonstrate analytically that the M1 contribution cancels. This is true for any ∆m = 2 transition, and one can see this in The ∆ recoils when it is produced by the proton ab- the plots in the Appendix, where every plot in the right sorbing the photon. The rotations of the moving state hand column has ∆m = 2 and no M1 contribution. are not given so simply as for a state at rest. It will turn 5 out that the corrections are not numerically large, but at time zero. The peaking of the wave function gives ~ we want to consider them. ~pf = k, and the angles for the outgoing ∆ that we might We find it helpful to consider the wave function of the call θ∆, φ∆ are the same a θk, φk. initial nucleon state. Even for situations that ultimately The Rarita-Schwinger spin-3/2 spinors are given in involve only plane waves, the manipulations of scatter- terms of spin-1 polarization vectors and spin-1/2 Dirac ing theory are justified by using wave packets for the spinors as states—see for example [23, 24]—and then taking limits. Here will use the twisted photon state as already given, X 3/2 1 1/2 but will more carefully consider the wave function of the u (~p,m) = (ˆp, λ)u(~p,µ). (21) µ m λ µ µ proton state. We want to localize it at the origin, but re- λ,µ alize that there is a playoff between localization and the uncertainty of the target momentum. We shall consider Given the structure of the Dirac spinor, one can define a that the uncertainties in both position and momentum two-spinor with a vector index are small judged by to scales over which other quantities in the calculation vary [23, 24]. X 3/2 1 1/2 χ (ˆp, m) = (ˆp, λ)χ(ˆp, µ) (22) The amplitude is µ m λ µ µ λ,µ 3 M = (d p)φei(~pi) ˆ where χ is a two-component spin-1/2 helicity state. 0 ~ × h∆(pf , mf )|H|N(~pi, mi), γ(κmγ kzΛb)i (15) Thence, where H is the interaction Hamiltonian. Inserting the ! 1 (E∆ + M∆) χµ(ˆp, m) expansion for the twisted photon gives uµ(~p,m) = √ , E∆ + M∆ ~σ · ~pχµ(ˆp, m) dφ ~ ~ k mγ imγ φk−ik·b M = A0 (−i) e φei(~pi) (23) ˆ 2π 0 ~ × h∆(pf , mf )|H(0)|N(~pi, mi), γ(k, Λ)i (16) where the ~σ are the 2 × 2 Pauli matrices. Under rotations, µ and χ transform simply using the where ~p = ~p + ~k. Further, f i Wigner functions d1 and d1/2, respectively, and with

dφk ~ ~ Wigner function theorems [25, 26] one can show that mγ i(mγ +mi)φk−ik·b ~ M = A0 (−i) e φei(~pf − k) ˆ 2π X 3/2 (pw) 1/2 −iφk 0 χµ(ˆp, m) = Rχµ(ˆz, m) = e dmm0 (θk)χµ(ˆz, m ), × Mm0 Λ dm m0 (θk) (17) i i i m0 0 0 (24) with mf = mi + Λ. This supposes that the wave function peaks sharply enough so that we can evaluate the plane wave amplitude at ~pi = 0, and do rotations on the with R = R(φk, θk, 0). This means that the upper com- nucleon as if it were at rest. There is only one Wigner ponents of the Rarita-Schwinger spinor will transform function, because we have not yet projected the ∆ onto under rotations with the same application of the Wigner states with spin quantized along the z-axis. functions as for a state at rest. This means that one 0 can do the same manipulations as for the no-recoil case, This is the amplitude for producing |∆(~pf , mf )i. The whole of the ∆ final state is and obtain results that look like the nonrelativistic case, 3 Eq. (11). This in turn means that although the final (d p ) ~ ~ X f 0 mγ i(mγ +mi)φk−ik·b ∆ is not a momentum eigenstate, one can express the |fi = |∆(~pf , mf )i (−i) e 0 ˆ 2Ef upper components, and hence the numerical bulk of the mf state, in terms of momentum eigenstate uµ spinors with ~ (pw) 1/2 × φei(~pf − k)M 0 d 0 (θk) (18) momenta in the z-direction. miΛ mimi However, for the lower components there is additional We get the coordinate wave function of this state using angular dependence in the ~σ · ~p term. In particular, the ∆ field operator and

0 0 −ipf ·x ˆ −iφk +iφk h0|Ψµ(x)|∆(~pf , mf )i = uµ(~pf , mf ) e , (19) ~σ · k = σ+ sin θke + σ− sin θke + σz cos θk, (25) leaving where σ± = σx ± iσy. One can still do the azimuthal 0 dφ uµ(~pf , m ) integral, but the extra φk dependence will bring in Bessel X k f mγ h0|Ψµ(x)|fi = (−i) functions with diﬀerent indices from the main term. 0 ˆ 2π 2Ef mf Doing the integral, and projecting onto the z-direction µ ~ ~ i(mγ +mi)φk−ik·b (pw) 1/2 spinoru ¯ (pz,ˆ mf ) leads to a main term that looks the × e φi(~x) M 0 d 0 (θk) (20) miΛ mimi same as the no-recoil result Eq. (11) plus a correction 6 term which we will have the temerity to write out, (a) mγ=2, Λ=1 M − E 0.010 (mi−mf ) i(mi+mγ −mf )φb ∆ ∆ X δM = A0 i e 2M∆ 0 mi 0.001 3/2 1 1/2 3/2 1 1/2 10-4 mf mf − 1/2 1/2 mf − 1 mf − 1/2 −1/2 ( A . U ) | 2 Total 3/2 | M 10-5 × sin θk d 0 (θk) M1 only mf −1,mi+Λ E2 only -6 3/2 1 1/2 3/2 1 1/2 10 + mf mf + 1/2 −1/2 mf + 1 mf + 1/2 1/2 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 b/λ 3/2 × sin θk d 0 (θk) (b) mf +1,mi+Λ mγ=2, Λ=1 3.0 3/2 + (cos θk − 1) d 0 (θk) mf ,mi+Λ 2.5 (pw) 1/2 × M 0 Jmf −mγ −mi (κb) d 0 (θk) . (26) Total mi,Λ mi,mi 2.0 M1 only

The nominal size of the correction term is pw 1.5 E2 only σ / 2 E − M k 1.0 ∆ ∆ = ≈ 1.9% , (27) 2M∆ 2M∆(E∆ + M∆) 0.5 as already noted in the introduction. 0.0 Implementing the corrections, the amplitude with pro- 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 b/λ jection onto the final ∆ state with mf = 3/2 is shown in Fig.3(c). The extra terms have some effect, as seen, 2 but very small. Corrections for other quantum number FIG. 4. (a) Matrix element squared, |M| , with summa- situations are visually even smaller. tion/averaging performed over baryon helicities. We choose the photon helicity Λ = 1 and total angular momentum projection mγ = 2. The blue solid line includes both M1 and E2 transitions, while the green dashed (black dotted) lines V. RATES, CROSS SECTIONS AND NOVEL show individual contributions from M1 (E2). (b) Same as SPIN EFFECTS (a) divided by the photon energy flux density obtainable from Eq. (6) and normalized to the plane-wave limit mγ = 1, Squaring the twisted amplitudes Eq.(11), summing θk → 0. over final helicities of ∆ and averaging over initial helicities of a nucleon, we obtain a quantity |M|2 that defines a transition rate for photoexcitation. The result is shown fixed the quantity l = m − Λ that corresponds to pho- in Fig.4a as a function of proton’s transverse position b γ γ ton’s orbital angular momentum (in a paraxial limit). with respect to the twisted photon’s axis. Namely, we form the following asymmetries: It can be seen that the contribution of the electric quadrupole amplitude E2 is almost independent of b, |M|2 − |M|2 whereas the magnetic dipole M1 contribution is sup- A(Λ) = Λ=1 Λ=−1 (28) M2 2 2 pressed in the vicinity of the photon vortex center as |M|Λ=1 + |M|Λ=−1 b → 0. Since this rate is obtained with a position- i.e., the beam spin asymmetry of transition rate and dependent photon flux, it is instructive to divide the rate by the flux, defining a position-dependent cross section (Λ) σΛ=1 − σΛ=−1 σ(b) of photoexcitation shown in Fig.4b. Aσ = , (29) σΛ=1 + σΛ=−1 The outcome shows a remarkable feature of the twisted photoexcitation: For E2 transitions the rate remains or the beam spin asymmetry of cross sections. Both nonzero as b → 0, while the flux turns to zero. It ef- these asymmetries would be identically zero for plane- fectively leads to an infinite cross section seen in Fig.4b wave photons if spatial parity is conserved. They may be at the vortex center. On the other hand, M1 absorption nonzero for the twisted photons because they arise due rate is proportional to the flux, same as for the cross- to the differences between twisted photoabsorption with section of plane-wave photoabsorption. aligned vs. anti-aligned spin (Λ) with respect to the or- Next, we compare absorption rates and cross sections bital angular momentum (lγ ). A similar effect for atomic for different values of photon helicity Λ, while keeping transitions was predicted theoretically in Ref.[27], with 7

(a) 1/b2 singularity near the vortex center exclusively due to lγ=1 ∗ the E2 amplitude GE. 0.0 VI. SUMMARY

-0.2 Total We have considered photoproduction of the ∆(1232) -0.4 M1 only baryon by twisted photons. The angular momentum se- 2

( Λ ) M E2 only

A lection rules for the transitions to definite final states -0.6 are different from what is possible with plane waves, and leads to additional opportunities. In particular, there are -0.8 transitions to which the main M1 component does not -1.0 contribute at all in the no-recoil or nonrelativistic limit, 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 and even with corrections implemented, contributes only b/λ very slightly. A measurement of these transitions can (b) isolate and measure the smaller E2 amplitude, which is lγ=1 a measure of the more complete structure of the baryon 1.0 Total states. M1 only Twisted photon experiments like these are clearly for 0.8 E2 only the future, as one needs to learn how to produce energetic

0.6 twisted photons, with large pitch angles and good control

( Λ ) σ over the location of the photon’s vortex axis relative to A 0.4 the target. However, there are good opportunities, and since gamma factories are under consideration, twisted 0.2 photon beams should also be considered.

0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 b/λ ACKNOWLEDGEMENTS

FIG. 5. Beam spin asymmetry for the transition rate (a) and A.A. thanks the US Army Research Office Grant flux-normalized cross section (b) as defined in Eqs.(28,29). W911NF-19-1-0022 for support and C.E.C. thanks the We took l = 1; notation for the curves is as in Fig.4. γ National Science Foundation (USA) for support under grant PHY-1812326.

E2 transitions shown to be responsible for cross section asymmetries. Appendix A: Larger collection of plots The results shown in Fig.5 demonstrate that the spin asymmetry of transition rate is dominated by M1 tran- Fig.6 presents a set of plots for twisted photon + pro- sition similarly to spin dependence of the photon flux, ton → ∆(1232), with varying choices for the twisted pho- but the asymmetry of cross section is dominated by E2 ton total angular momentum m , photon helicity Λ, and contribution near the photon vortex center. γ final ∆ spin projection mf . The vertical axis shows the The differences in contributions of E2 and M1 transi- amplitude magnitudes, and the horizontal axis shows the tions in the twisted photoabsorption can be better under- impact parameter or the offset between the target loca- stood from comparing analytic expressions if we perform tion and the photon vortex axis. a Taylor expansion of the cross section near the vortex All plots have the proton polarized with m = −1/2. center in a paraxial limit (c.f. Eqs.(21-24) of Ref.[12]): i Plots with mi = +1/2 are identical if one uses parity to 3G∗2 change the other quantum numbers appropriately. Each σ ∝ 3G∗2 + G∗2 + E . (30) (mγ =2,Λ=1) E M 2 row has mγ and Λ, the helicity of the incoming photons (bπ/λ) in a Fourier decomposition, labeled. In each row the spin Comparing the expression above with the plane-wave projection of the final ∆ goes from −3/2 on the left to cross section +3/2 on the right. These plots include the recoil corrections. One no- pw ∗2 ∗2 σ ∝ 3GE + GM , (31) tices that all plots in the right-hand column show only small contribution from the elsewhere dominant M1 am- it becomes apparent that the twisted cross section has a plitude. 8

[1] U. Jentschura and V. Serbo, Eur. Phys. J. C 71, 1571 [13] A. M. Yao and M. J. Padgett, Advances in Optics and (2011), arXiv:1101.1206 [physics.acc-ph]. Photonics 3, 161 (2011). [2] V. Petrillo, G. Dattoli, I. Drebot, and F. Nguyen, Phys. [14] K. Y. Bliokh and F. Nori, Phys. Rept. 592, 1 (2015), Rev. Lett. 117, 123903 (2016). arXiv:1504.03113 [physics.optics]. [3] D. Budker, J. R. Crespo López-Urrutia, A. Dere- [15] U. Jentschura and V. Serbo, Phys. Rev. Lett. 106, 013001 vianko, V. V. Flambaum, M. W. Krasny, A. Petrenko, (2011). S. Pustelny, A. Surzhykov, V. A. Yerokhin, and M. Zolo- [16] A. Afanasev, C. E. Carlson, and A. Mukherjee, Phys. torev, Annalen der Physik 532, 2000204. Rev. A 88, 033841 (2013). [4] P. Liu, J. Yan, A. Afanasev, S. V. Benson, H. Hao, S. F. [17] G. F. Quinteiro, C. T. Schmiegelow, D. E. Reiter, and Mikhailov, V. G. Popov, and Y. K. Wu, “Orbital angular T. Kuhn, Phys. Rev. A 99, 023845 (2019). momentum beam generation using a free-electron laser [18] H. M. Scholz-Marggraf, S. Fritzsche, V. G. Serbo, oscillator,” (2020), arXiv:2007.15723 [physics.acc-ph]. A. Afanasev, and A. Surzhykov, Phys. Rev. A 90, 013425 [5] M. Drechsler, S. Wolf, C. T. Schmiegelow, and (2014). F. Schmidt-Kaler, arXiv e-prints , arXiv:2104.07095 [19] G. C. Wick, Annals Phys. 18, 65 (1962). (2021), arXiv:2104.07095 [quant-ph]. [20] H. F. Jones and M. D. Scadron, Annals Phys. 81, 1 [6] C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, (1973). U. G. Poschinger, and F. Schmidt-Kaler, Nature Com- [21] Eq. (13) was obtained using Ref. [9] and agrees with [20] if munications 7, 12998 (2016). one identifies H(0) here with −ieΓ there and also inserts [7] A. Afanasev, C. E. Carlson, C. T. Schmiegelow, J. Schulz, the isospin factor. F. Schmidt-Kaler, and M. Solyanik, New Journal of [22] A. Afanasev, C. E. Carlson, and A. Mukherjee, Journal Physics 20, 023032 (2018). of the Optical Society of America B Optical Physics 31, [8] M. Solyanik-Gorgone, A. Afanasev, C. E. Carlson, C. T. 2721 (2014). Schmiegelow, and F. Schmidt-Kaler, J. Opt. Soc. Am. [23] J. R. Taylor, Scattering Theory: The Quantum Theory of B 36, 565 (2019). Nonrelativistic Collisions (Dover, Mineola (2006)), (orig- [9] V. Pascalutsa, M. Vanderhaeghen, and S. N. Yang, Phys. inally Wiley, New York (1972) and also Krieger, Malabar, Rept. 437, 125 (2007), arXiv:hep-ph/0609004. Florida (1983)); particularly see chapters 2 and 3. [10] P. A. Zyla et al. (Particle Data Group), PTEP 2020, [24] M. E. Peskin and D. V. Schroeder, An Introduction to 083C01 (2020). quantum field theory (Addison-Wesley, Reading, USA, [11] P. Rynkun, P. Jönsson,G. Gaigalas, and C. F. Fischer, 1995), particularly section 4.5. Atomic Data and Nuclear Data Tables 98, 481 (2012), [25] M. E. Rose, Elementary Theory of Angular Momentum 2 2 (The specific transition is P1/2 → D3/2 (142 nm) on p. (John Wiley and Sons, New York, 1957). 501 of this article. Both states have 3 electrons in a 2s2p2 [26] A. R. Edmonds, Angular Momentum in Quantum Me- configuration. The atom in question is Ne VI, which has chanics (Princeton University Press, Princeton, NJ, 5 electrons, the same as neutral Boron.). 1957). [12] A. Afanasev, C. E. Carlson, and M. Solyanik, Phys. Rev. [27] A. Afanasev, C. E. Carlson, and M. Solyanik, Journal of A 97, 023422 (2018). Optics 19, 105401 (2017). In[3557]:=

Twisted_gN->Delta4.nb 11

For these plots, mi = -1/2; k (photon beam momentum) =

0.339704 GeV (λ = 3.64977 fm); θk = 0.2 radians. Amplitude units are arbitrary, but the same for each plot.

* * * * Solid curves have both GM and GE with GE /GM = 3. % * * ; green dashed curve is GM alone, black dotted curve is GE alone. 9

mγ = -2, Λ = -1, mi = -1/2, mf = -3/2 to +3/2 80 4 0.07 | 600 0.06 60 3 0.05 400 0.04 40 2 0.03 Out[3575]= 200 20 1 0.02 Amplitude

| 0.01 0 0 0 0.00 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 b/λ b/λ b/λ b/λ

mγ = -1, Λ = -1, mi = -1/2, mf = -3/2 to +3/2

| 1200 1000 80 4 0.06 60 3 800 0.04 600 40 2 Out[3580]= 400 0.02

Amplitude 20 1

| 200 0 0 0 0.00 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 b/λ b/λ b/λ b/λ

mγ = 0, Λ = -1, mi = -1/2, mf = -3/2 to +3/2 0.08 | 150 5 600 4 0.06 100 400 3 0.04 50 2 Out[3585]= 200 0.02 Amplitude

| 1 0 0 0 0.00 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 b/λ b/λ b/λ b/λ

mγ = 0, Λ = 1, mi = -1/2, mf = -3/2 to +3/2 150 400 8 | 6 100 300 6 4 200 4 50 Out[3591]= 2 100 2 Amplitude | 0 0 0 0 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 b/λ b/λ b/λ b/λ

mγ = 1, Λ = 1, mi = -1/2, mf = -3/2 to +3/2 6 80 700 | 5 600 8 4 60 500 400 6 3 40 300 4 Out[3596]= 2 200

Amplitude 20 2 | 1 100 0 0 0 0 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 b/λ b/λ b/λ b/λ

mγ = 2, Λ = 1, mi = -1/2, mf = -3/2 to +3/2 70 400 | 5 60 15 4 50 300 40 10 3 200 2 30 Out[3602]= 20 100 5 Amplitude | 1 10 0 0 0 0 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 b/λ b/λ b/λ b/λ

FIG. 6. Plots of amplitudes for γ + p → ∆ with twisted photons. The protons all have spin projection mi = −1/2. The mγ and Λ for each row are labeled, and the spin projections of the ﬁnal ∆ run from −3/2 to +3/2. left to right. The units of the amplitude are arbitrary, but are the same for each plot. The pitch angle for each plot is θk = 0.2. The red (for mγ = ∆m) or ∗ ∗ blue (otherwise) solid curves are the total; the green dashed curves are GM alone and the black dotted curves are GE alone.