Baryonic dark forces in electron-beam fixed-target experiments

New GeV-scale dark forces coupling predominantly to quarks offer novel signatures that can be produced directly and searched for at high-luminosity colliders. We compute the photon-proton and electron-proton cross sections for producing a GeV-scale gauge boson arising from a $U(1)_B$ gauge symmetry. Our calculation relies on vector meson dominance and a phenomenological model for diffractive scattering used for vector-meson photoproduction. The parameters of our phenomenological model are fixed by performing a Markov Chain Monte Carlo fit to existing exclusive photoproduction data for $\omega$ and $\phi$ mesons. Our approach can be generalized to other GeV-scale dark gauge forces.


I. INTRODUCTION
Luminosity-frontier experiments have a unique niche for discovering new gauge forces that are light and weakly-coupled to the Standard Model (SM).These searches have been motivated in part by the muon (g − 2) anomaly [1][2][3], the ATOMKI anomalies [4,5], and dark matter [1,[6][7][8][9][10].Irrespective of particular anomalies, however, it is important to explore all possible ways new forces may couple to the SM.Most searches have focused on a dark photon arising via kinetic mixing [11].Since the dark photon couples to the electromagnetic current, many experiments rely on its leptonic couplings to search for e + e − or µ + µ − resonances.Alternative approaches are also being pursued in case dark photons decay invisibly to light dark matter.(See [12][13][14] for reviews.) Different search strategies are needed to discover a leptophobic gauge force that couples predominantly to quarks.The simplest model is the B boson that arises from a gauged U (1) B baryon number symmetry [15][16][17][18][19][20][21][22][23][24].The interaction Lagrangian is where g B is the U (1) B gauge coupling.We also include dark-photon-like couplings where ε is the kinetic-mixing parameter, e is the usual proton electric charge, and Q q denotes quark electric charges in units of e.Even if ε = 0 at tree-level, a nonzero value arises via radiative corrections from heavy quarks [21].Depending on ε, the B boson is subject to many constraints from dark photon searches [25] and flavor physics [26,27].Since these effects are somewhat modeldependent [28], here we focus only on the leptophobic coupling g B , i.e., taking the limit g B ≫ εe.
In this case, the dominant B-boson decays are for m B in the ranges 140−620 MeV and 620 MeV−1 GeV, respectively [29].The subleading decay B → π + π − is forbidden by G-parity but can proceed via isospin-breaking, e.g., due to ρ-ω mixing.When m B < m π , we expect B → e + e − via the small but nonzero kinetic mixing.One discovery strategy is searching for leptophobic gauge bosons in light meson decays at meson factories.For the B boson, the two most promising channels are η → Bγ → π 0 γγ , ϕ → ηB → ηπ 0 γ . ( These processes mimic rare decays in the SM [17,29] and are search targets for the Jefferson Eta Factory [30] and KLOE-2 [31,32].Belle has also searched for and found a null result for η → Bγ → π + π − γ [33].However, the disadvantage of these searches is that the B-boson mass reach is limited by available phase space from the parent meson, ∼ 500 MeV.Above this range, dijet decays in heavy-flavor quarkonia provide the strongest constraints [22].An alternative strategy is searching for new gauge bosons produced directly in collisions.Naturally, the advantage is that more phase space is available (depending on beam energy) and more massive gauge bosons can be searched for.In this work, we calculate the B-boson cross section via real and virtual photoproduction γ ( * ) p → Bp , which can be searched for as a narrow π 0 γ or π + π − π 0 resonance in electron-proton collisions.
(For simplicity, we consider a proton target only.) The high-luminosity frontier for electron-proton collisions is of great interest for our understanding of strong dynamics and the structure of hadrons.This includes fixed-target experiments, e.g., the current 12-GeV Continuous Electron Beam Accelerator Facility (CEBAF) at Jefferson Laboratory [34] and its proposed 22-GeV upgrade [35], as well as various proposed electronhadron colliders [36][37][38].Here we focus on electron-beam fixed-target experiments and their potential for uncovering new leptophobic gauge bosons hidden in QCD.
Unfortunately, it is not possible to calculate photoproduction from first principles.At large center-of-mass energies, √ s ≫ GeV, photon-hadron interactions share the same soft diffractive behavior as purely hadronic interactions: namely, cross sections that grow weakly with √ s and are dominated by small momentum transfer, |t| ≲ GeV [39]. 1 Thus, we are led to a phenomenological model based on two assumptions.First, the similarity between photon-hadron and hadron-hadron collisions points to the secretly hadronic nature of the photon.This is welldescribed by vector meson dominance (VMD), which assumes that external gauge fields couple to hadrons by mixing with light vector mesons (see e.g.[39,41]).Second, the common scaling behavior observed in different hadronic processes points to a universal model to describe diffractive scattering.This is well-described by the soft pomeron model [42].
We expect B-boson physics to follow similarly, provided m B ≲ GeV.Here we construct a phenomenological model for (4) based on the same physics used in the literature for describing photon-proton collisions in the Standard Model.Our model has many phenomenological parameters, some of which we obtain from the literature.The remainder we determine by calculating the photoproduction cross sections for ω, ϕ mesons within our model and fitting them to experimental data.Our fit spans center-of-mass energies √ s ∼ 2 − 94 GeV, covering the full range of presently available data except for the threshold region ( √ s ∼ 1 − 2 GeV).Ultimately, we are able to predict the B-boson photoproduction cross section solely in terms of the new physics parameters m B and α B = g 2 B /(4π).The methods used can be adapted to other leptophobic gauge models as well, which we briefly discuss at the end.
Finally, we note related work by Fanelli and Williams [43] that previously calculated the Bboson photoproduction cross section at fixed-target experiments.Their calculation and ours both rely on VMD to model B-boson couplings to hadrons.However, they parametrize the remaining part of the amplitude simply in terms of SM cross sections for γp → ωp, ϕp, whereas our work provides a direct calculation that is based on current theoretical models in the literature and calibrated to existing photoproduction data.We also consider virtual photons in the initial state, which is needed to describe B-boson electroproduction.
The remainder of this work is organized as follows.Sec.II describes the phenomenological model used in our calculation.Sec.III gives the results of our fit to ω, ϕ-meson photoproduction data, used to fit unknown parameters in our model.Secs.IV and V present our results for B-bosons produced via real and virtual photoproduction, respectively.We also provide a comparison to the results of Ref. [43].Our conclusions follow.Additional material is provided in the appendices.
Analytic formulae in this work have been coded up in Python and are available at https: //github.com/dark-physics/baryonic-dark-forces.

II. PHOTOPRODUCTION MODEL
To calculate the photoproduction cross section for the B boson, we first use VMD to expand the matrix element as in terms of SM matrix elements for vector-meson photoproduction.The coefficients c V are calculated below and neglecting isospin-violating effects (both in the SM and due to kinetic mixing) we need only consider ω, ϕ mesons.Next, we calculate the SM matrix elements for vector-meson photoproduction based on a phenomenological model, following previous literature [44][45][46][47][48][49][50].For ω-photoproduction, it is longknown that the cross section is dominated by one-pion-exchange at low energies and diffractive scattering at high energies [44,46].For ϕ-photoproduction, meson exchange is suppressed and diffractive scattering dominates [51,52].In our setup, we take a t-channel model that includes exchange of both light pseudoscalars (π 0 , η) and the pomeron.
It is customary to decompose the total matrix element as a sum of natural (+) and unnatural (−) parity amplitudes, i.e., which in our setup arise, respectively, from pomeron (+) and pseudoscalar (−) exchange.These contributions can be separately extracted from photoproduction with polarized photons [53].Here we consider the unpolarized differential cross section where The two contributions do not interfere with one another in the limit √ s ≫ m p .Analogous formulae hold for γp → Bp as well.Fig. 1 shows the Feynman diagrams for B-boson photoproduction in our t-channel model.The left diagram is the natural-parity amplitude from pomeron exchange (jagged line), while the right diagram is the unnatural-parity amplitude from pseudoscalar-meson exchange (dashed line).Black dots denote phenomenological vertices in our model, which we determine from experimental data.Crossed boxes denote mixing between vector mesons and external gauge fields, a la VMD.Without VMD mixing with the B boson, the same diagrams and interactions correspond to ω, ϕ-meson photoproduction, which we also calculate.In the remainder of this section, we go through the calculations in detail.

A. Vector meson dominance
Though VMD is well-known [39,41], we summarize the basic idea here for completeness and to fix our notation.For example, an amplitude with an external photon, with momentum q and polarization λ, can be expressed as a sum over similar diagrams where the photon couples by mixing with a vector meson V : The V -γ mixing part of the diagram introduces the factor where f V is the meson decay constant and F V (q 2 ) is a form factor, normalized to F V (0) = 1 for an on-shell photon.For ω and ϕ mesons, it suffices to work in the narrow-width approximation, taking a simple Breit-Wigner form factor For the ρ 0 meson, its large width necessitates a more complicated form.
Additionally, T V is the U (3) flavor generator for a given meson and 3 ) is the electric charge operator.For ideal mixing, the lightest vector mesons that mix with the photon have generators The decay constant f V is defined by where ε µ is the polarization vector, following [54].These are extracted from the measured partial widths Γ(V → e + e − ) [40] according to The values are listed in Table I.
For a B boson (assumed to be on-shell), the VMD mixing factor is where I is the identity matrix.For εe ≪ g B , B bosons are produced predominantly via isoscalar ω, ϕ mesons.Accordingly, the coefficients in Eq. ( 5) are The B-boson photoproduction matrix elements are related to those for ω, ϕ mesons by the formula
The pseudoscalar-photon-vector-meson couplings g φγV are determined precisely from the measured partial widths Γ(V → φγ) [40], according to the formula Using U (3) flavor symmetry, we assume g ργπ , g ργη , g ωγπ , g ωγη , and g ϕγη have the same relative sign.The final coupling g ϕγπ arises through ω-ϕ mixing away from the ideal limit.Parametrized as g ϕγπ = − tan δ V g ωγπ , Ref. [58] performed a global fit to meson decays and found an angle δ V ≈ −3 • relative to the ideal limit.Hence, we take g ϕγπ to have the same relative sign as the others.The numerical values of these couplings are in Table III.
The matrix element for vector-meson photoproduction is Here u (ū) is the incoming (outgoing) proton spinor, and Each vertex in Eq. ( 16) is additionally dressed with a form factor where Λ represents a cutoff scale [47].From Eq. ( 18), it is straightforward to compute the differential cross section in the limit √ s ≫ m p .
For B boson photoproduction, we need consider processes involving only ω, ϕ-mesons.Hence, we need six cutoffs in Eq. ( 20) These are determined from photoproduction data.This has been done for ω photoproduction and the first four cutoffs in (22) were determined to be in the range 0.5 − 1 GeV [49,60].Here we determine all six cutoffs through a joint fit to ω and ϕ photoproduction data, described below.
With the model parameters fixed, we calculate the unnatural parity contribution to the B-boson cross section following Eq.( 15):
Following Ref. [50], the matrix element is On the right-hand side of Eq. ( 24), the first line represents the pomeron-nucleon vertex, which is parametrized by β PN N F 1 (t), where β PN N is the coupling constant and F 1 (t) is the Dirac form factor of the proton [42].The second line represents the pomeron-vector-meson vertex, parametrized by coupling constants a PV V , b PV V , as well as a (common) meson form factor F M (t).We also have the tensors [50] The third line of Eq. ( 24) represents the pomeron propagator, where the pomeron trajectory is Fits to pp and pp scattering data have determined [65] In contrast, the pomeron-vector-meson couplings are less well-known.First, we expect a Pωω ≈ a Pρρ and b Pωω ≈ b Pρρ [66].Second, it is argued that the total (inclusive) cross section for ρp scattering is related to that of π ± p scattering, which holds to a good approximation experimentally [50].This yields a relation (and similarly for the ω couplings).A similar argument relating ϕp and K ± p scattering yields [67] Here we do not impose Eqs. ( 29) or (30), but rather we impose weaker priors which are discussed in Appendix A.
Next, we use VMD to calculate the photoproduction matrix element from Eq. ( 24): where q 2 = 0 for an on-shell photon. 3Retaining only the leading term in powers of s, the vectormeson photoproduction cross section is where pV (t) is a joint form factor for both pomeron-proton and pomeron-vector-meson interactions.In Eq. ( 24), this is However, here we adopt a different ansatz with a different slope B V for each vector meson [63], which provides a reasonably good fit to our dataset.
For B-boson production, only ω, ϕ-processes contribute.The matrix element, following from Eq. ( 15), is where the relative minus between the two terms is fixed by VMD since the ω and ϕ amplitudes are proportional to This reflects the fact that the pomeron-exchange amplitude for γp → Bp must vanish in the SU (3)-flavor-symmetric limit, since the photon and B boson couple to orthogonal generators.Next, it is helpful to introduce the following general linear combinations of couplings with coefficients In terms of these quantities, the natural-parity contribution to the cross section is which is evaluated as a function of α B and m B once the parameters of the phenomenological model are fixed.
In the present work, we treat the parameters as freely varying in our fit.However, we keep β PN N and s 0 as fixed in Eq. ( 28).While we do not expect uncertainties in these latter parameters to be small, variation can be (to some extent) absorbed into the other parameters.We defer a joint fit to both nucleon scattering and photoproduction to future work.

III. NUMERICAL FIT
To determine the B-boson photoproduction cross section, we perform a fit to experimental data to determine the phenomenological parameters entering our model.Our dataset consists of differential cross section measurements for exclusive ω, ϕ-photoproduction.These include high precision measurements with the CEBAF Large Acceptance Spectrometer (CLAS) [69,70], which go from threshold up to √ s ≈ 2.8 GeV; older measurements [71][72][73], which extend up to √ s ≈ 20 GeV; and from ZEUS at much larger energies, √ s ≈ 70 − 94 GeV [74][75][76].This latter is particularly important for determining the pomeron contribution in γp → ωp, which is not wellconstrained from low-energy data alone.We restrict our dataset to lie in range since our model aims to give the leading contribution in the diffractive limit.Outside this range, diffractive scattering and/or meson-exchange are no longer dominant [60,70] and subdominant processes neglected in our model can become important, e.g., σ-meson exchange [45], nucleon excitations [48,49], or exchanges of additional reggeons [50].Since our aim is to describe the leading contributions to B-boson production, it suffices to neglect these.We also include a 10% systematic error on all data points, added in quadrature with statistical errors.
In addition, we include data for the total photon-proton cross section measured by H1 [77] and ZEUS [78] at √ s ≈ 200 GeV.Discussed in Appendix A, this data is also important for constraining the ω-pomeron coupling.
Our fit has a similar but complementary spirit to other photoproduction fits from previous literature [49,60,68].The models adopted therein include many additional contributions needed to describe ω and ϕ photoproduction data across the full kinematic ranges in t, however, these fits are each limited to smaller ranges of √ s.For new gauge forces, a complete phenomenological model including all known contributions would be ultimately desirable, but we defer this to future work.Fixed parameters are given in Tables I and II.Here we take the central values as input and do not propagate uncertainties in our analysis.Other parameters are given in Table III.Here β PN N , α ′ P (0) are fixed from pp, pp scattering data [65].We fit the remaining fifteen parameters from experimental data using a Markov Chain Monte Carlo analysis.For some parameters, we adopt Gaussian priors to exclude values far from expectations (discussed in Sec.II).
The results from our fit are given in Table III.The fitted parameters quoted are medians and one-sigma intervals, except for a Pωω which is consistent with zero and a one-sigma upper limit is provided.For the most part, our results are consistent with values in the literature.A similar phenomenological fit to ω data from CLAS [60] yielded the following values albeit with different assumptions for the pseudoscalar-nucleon couplings and pomeron-exchange amplitude, which are in agreement with our results.The pomeron trajectory intercept found in our fit is in good agreement with the value α P (0) = 1 + ϵ P = 1.0808 quoted in the literature, as extracted from pp and pp scattering [65].Our vector-meson-pomeron couplings satisfy the following relations to be compared with Eqs. ( 29) and (30).
To illustrate the results of our fit, Fig. 2 shows the current world dataset for exclusive ω, ϕphotoproduction cross sections as a function of √ s (data points).Filled points represent datasets that were included in our fit, while open points were not, either because differential cross section data was not publicly available or because √ s was near the threshold region.The shaded bands are the results from our phenomenological model, where the band width represents a 90% confidence interval in our fitted parameters.
Our model for ω-photoproduction appears in good agreement with data.However, our model for ϕ-photoproduction appears to systematically under-predict experimentally-determined cross sections by O(30%).We emphasize that our model is fit to differential cross section data, whereas determining the total cross section both on the theory and experimental sides requires extrapolating the differential cross section to the forward-angle limit, which may lead to additional systematic uncertainties.In Appendix B, we provide a comparison between our model and experimental data for the differential photoproduction cross section included in our fit.parameter prior fit value parameter prior fit value TABLE III.Summary of fixed and fitted parameters in our model.Uncertainties for fixed inputs are quoted where known, but are not propagated in our analysis.All hadron masses and widths, and α em , are also fixed and are taken from [59].Fitted parameters are determined in our fit to experimental data, subject to a Gaussian prior.The quoted best-fit values include statistical uncertainties only.

IV. B-BOSON PHOTOPRODUCTION
The differential cross section for B-boson photoproduction is The unnatural (−) and natural (+) parity contributions are given by Eqs. ( 23) and ( 39), respectively.In Fig. 3, we plot the B-boson photoproduction cross section as a function of √ s, for various masses m B .The left panel shows the total cross section, while the center and right panels show the unnatural-and natural-parity cross sections separately from pseudoscalar and pomeron exchange, respectively.For the parameter range shown, the cross section tends to be dominated the pomeron contribution which is only weakly-dependent on √ s, characteristic of diffractive scattering, except for near threshold where pseudoscalar-exchange is important. 4he behavior of the cross section with B boson mass is shown in Fig. 4.Here we show the photoproduction cross section relative to that for ω-mesons, which in our model is σ(γp → ωp) ≈ 1.2 µbarn for √ s = 4 GeV The darker (blue) band shows the prediction from our phenomenological model.Due to vector-meson mixing, the cross section is strongly enhanced for m B near m ω or m ϕ .Next, we compare our results to those of Fanelli and Williams [43].Their formula is where the phase space factors are approximately Φ(m 2 B )/Φ(m 2 ω,ϕ ) ≈ 1 in the large-s limit.Similar to our work, Eq. ( 45) treats B production using VMD via ω and ϕ mixing, with the added approximation 2m 2 ω /f 2 ω ≈ 2m 2 ϕ /f 2 ϕ ≈ 12π (which is true to better than 20%).The matrix elements arising from ω or ϕ mixing are parametrized simply in terms of the respective SM cross sections, except for unknown relative phases φ ± .
Taking inputs from Ref. [43], the results of Eq. ( 45) are shown in Fig. 4 (light gray band).The phase φ + is allowed to vary between 0 and π, which sets the width of the band, while φ − does not enter Eq. ( 45) since σ − (γp → ϕp) is neglected.The unknown phase φ + limits the precision of Eq. ( 45) in the absence of additional input (as the authors discuss).On the other hand, our predictions are more precise and have no unknown phase, as it is fixed by the relative sign in Eq. (34).That is, we have cos φ + = −1 for m B < m ω or m B > m ϕ , while cos φ + = +1 for m ω < m B < m ϕ .Fixing this sign in Eq. ( 45) shows good agreement with our work.

V. B-BOSON ELECTROPRODUCTION
A. Theoretical preliminaries B bosons can be produced in electron-proton collisions, shown in Fig. 5, analogous to virtual Compton scattering.The particles involved have the following four-momenta: k (k ′ ) and p (p ′ ) for the incoming (outgoing) electron and proton, respectively, q for the photon, and q ′ for the B boson.The usual kinematic variables for deep inelastic scattering are where Q 2 is the momentum transfer, and y is the fractional electron energy loss and ν is the photon energy in the lab frame (for a fixed proton target).The invariant mass-squared of the photon-proton system is W 2 , which was defined as s previously in Sec.II.Here we denote s tot = (p + k) 2 as the total center-of-mass energy-squared for the entire electron-proton system, while t = (q − q ′ ) 2 is the same as previously defined for photoproduction.Lastly, it is useful to note the following relations and For unpolarized scattering and fixed beam energy, there are three independent kinematic variables, which we take to be y, Q 2 , t.The triple-differential cross section for electroproduction can be expressed as It is customary to follow Hand's convention [93] to express the electron-photon part of the cross section (following standard quantum electrodynamics) as an effective flux of transverse (T ) and longitudinal (L) virtual photons which multiply the corresponding photoproduction cross sections, in the limit W ≫ m p . 5 Also, we have Q 2 min = m 2 e y 2 /(1 − y).Before we proceed further, let us provide the virtual-photoproduction cross sections for vector mesons in our model.These formulae are not needed here, but are included for completeness and may be used in future work for constraining our model with vector-meson electroproduction data away from the Q 2 = 0 limit.First, we write the differential cross sections as a sum of natural and unnatural parity contributions Under Hand's convention, the differential cross section is evaluated as where the photon polarization λ is excluded from the sum over spins.Next, following Sec.II, we evaluate the squared matrix elements in Eqs. ( 18) and (24).Here, however, the definite photon polarization vector ϵ µ λ enters explicitly and we are left with various Lorentz scalar products involving ϵ µ λ .To proceed, we work the frame where the initial proton is at rest and the virtual photon momentum is aligned along the z-axis, i.e., p µ = (m p , 0, 0, 0) , q µ = ν, 0, 0, The transverse and longitudinal polarizations take the form, respectively, satisfying (for space-like photon momentum q µ ) 5 Following the standard formula [94], the cross section for a virtual photon scattering on a proton at rest into a generic final state X, labeled by momenta k i , is Under Hand's convention, the photon momentum |q| in the flux prefactor is replaced by ν − Q 2 /(2m p ), which is the three-momentum for an equivalent real photon that would give the same total center-of-mass energy W as the virtual photon.The electroproduction cross section does not depend on this choice provided different kinematic factors are absorbed into the definitions of the effective fluxes Γ T,L [39].
For the longitudinal cross section dσ (±) L (γp → V p)/dt, the Lorentz scalar products needed are taking the limit W ≫ m p , as well as ϵ 0 • q = 0 and ϵ 0 • p ′ = ϵ 0 • (p − q ′ ).For the transverse cross section, we average over transverse polarizations The identity is useful to express the sum over transverse polarizations in Eq. ( 59) in terms of Lorentz scalar products given above.With these manipulations, the positive-parity cross sections from pomeron exchange are dσ The negative-parity cross sections from pseudoscalar exchange are We retain only the leading terms in powers of W , assuming W ≫ m p , m B , |t|.(We do not assume that Q 2 is small compared to W 2 .)In the Q 2 = 0 limit, the transverse cross sections reduce to those for photoproduction given above, while the longitudinal cross sections vanish as expected.
Next, we turn to B-boson virtual-photoproduction.As above, the cross section is a sum of natural and unnatural parity contributions Using the Feynman rules of our model and the same manipulations given above, we have the following results.The pseudoscalar-exchange contributions are where The pomeron-exchange contributions are where a PγB , b PγB are defined in Eqs.(36) and (37) and we take the limit W ≫ |t|, m p .Again, for Q 2 = 0, the transverse cross sections reduce to our previous results for real photoproduction given in Eqs. ( 23) and (39), while the longitudinal cross sections vanish as expected.

B. Results
Figure 6 shows our predictions for the virtual-photoproduction cross sections for B bosons, obtained by integrating the above formulas over t.The dark band (solid lines) is the transverse cross section, while the lighter band (dashed lines) is the longitudinal cross section, as a function of Q 2 .Panels correspond to a grid of values for m B (rows) and W (columns).The width of each band represents the 90% confidence interval from our parameter fit.
For Q 2 < 0.1 GeV 2 , the effect of photon virtuality is neglible.The transverse cross section is asymptotically equal to the real-photoproduction cross section, while the longitudinal cross section is comparatively suppressed.In this regime, the electroproduction cross section is precisely determined by our phenomenological fit.On the other hand, for larger Q 2 , the longitudinal cross section may become comparable to the transverse one.In this case, our fit does not well-constrain model predictions.Presumably, this could be improved by including data from vector-meson electroproduction at larger Q 2 in our fits, but this remains for future work.
In Fig. 7, we show the total B-boson electroproduction cross section on a proton target integrated over the kinematically-allowed region, for electron beam energies E beam = 11 GeV and 22 GeV on a fixed proton target.This is representative of current and future beam energies at Jefferson Laboratory [34,35].Similar to B-boson photoproduction, our predictions vary most with m B , but are less sensitive to the center-of-mass energy.For fixed m B , our predictions have relatively small uncertainties since the integral is dominated by the real-photoproduction region with small-Q 2 , i.e., the diffractive region where our model is directly constrained by experiment.
For comparison, beam luminosities with the 22-GeV upgrade at Jefferson Laboratory are projected to be L ∼ (10 35 −10 38 ) cm −2 s −1 [35].According to Fig. 7, the total electroproduction cross section is in range ∼ (0.03 − 300) × α B µbarn.This translates into a total B-boson production rate of O(30 − 300, 000 Here, we scale our predictions relative to α B = 0.01, which is the approximate upper limit appli- cable for most of B-boson parameter space in the ∼ 0.5 − 1 GeV mass range [28].
Of course, acceptance rates in real experiments are less than production rates due to incomplete coverage of kinematic phase space, i.e., from detector limitations or selection cuts.Along these lines, Fig. 8 shows the kinematic distributions of several quantities in B-boson electroproduction on a fixed proton target: scattering angle θ and relative energy of the scattered electron in the lab frame, and the center-of-mass energy of the photon-proton system relative to that of the total electron-proton system.The vertical axis in these plots represents the marginalized probability distributions in these variables, with the total integral under the curves normalized to unity.It is clear that the process is dominated by forward-scattering where the photon-proton system represented in Fig. 1 is near threshold.

VI. CONCLUSIONS
Leptophobic gauge forces can provide different signatures compared to dark photons and other new states coupled to leptons.As many experiments continue the search for new physics at the GeV scale, it is important to consider all possibilities.The B boson is the minimal model along these lines and is currently being searched for in rare meson decays.However, direct production of GeV-scale B bosons in colliders offers a complementary strategy that has not yet been searched for.
In this work, we calculated the real and virtual photoproduction cross sections for B bosons on a proton target.Combined with knowledge of B-boson decay channels [29], these results can be used in experimental searches to make predictions and (in the absence of a discovery) set limits on B boson parameter space.Our calculation is based on phenomenological models for diffractive vector-meson photoproduction in the SM, as well as VMD.We performed a comprehensive fit to the current world's dataset for ω-meson and ϕ-meson photoproduction to fix the phenomenological parameters of our model.Our formulae contain complete kinematic information and can be used to determine acceptance efficiencies in experiments.Our phenomenological approach can be generalized to other leptophobic Z ′ models as well.For iso-singlet Z ′ bosons, this is straightforward modification of Eq. (15).However, if the Z ′ boson is not an iso-singlet, one must consider ρ 0 -meson mixing in addition to ω, ϕ-mixing considered here.In this case, one must extend the t-channel model to include an additional state, e.g., σ-meson exchange, in order to reproduce experimental data for ρ 0 -meson photoproduction [45].Extending our model along these lines and fitting to experimental data would be desirable, but we leave this to future work.
Next, we calculate the total cross section for photon-proton scattering.Using the optical theorem and VMD, we have setting q 2 = 0 for a real photon.The total photon-proton cross section has been measured by the H1 [77] and ZEUS [78] collaborations to be where the uncertainties represent statistical and systematic errors, respectively.Here we impose these measurements as an additional constraint on our model via Eq.(A9), with the added assumption b Pρρ = b Pωω .

FIG. 3 .
FIG. 3. Shaded bands show the B-boson photoproduction cross section from our model (90% confidence intervals), as a function of center-of-mass energy √ s and for different masses m B .Cross sections scale linearly with α B and have been normalized to α B = 1.

FIG. 4 .
FIG.4.Dark (purple) shaded band shows total cross section for B-boson photoproduction, relative to that for ω mesons, from our phenomenological model (90% confidence intervals).Light (gray) shaded band shows same quantity computed by Fanelli and Williams[43], where width comes from varying the unknown phase φ + in the range [0, π].Center-of-mass energy √ s = 4 GeV and coupling α B = 1 are fixed.

1 Q 2 (
FIG.6.B-boson photoproduction cross sections for transverse (darker band, solid lines) and longitudinal (lighter band, dashed lines) virtual photons are shown as a function of Q 2 for a grid of m B (rows) and W (columns) values.Bands denote predictions from our phenomenogical model (90% confidence intervals).Coupling α B = 1 is fixed.

FIG. 7 .
FIG. 7. Total B-boson electroproduction cross section is shown as a function of m B for electron beam energy E beam = 11 GeV and 22 GeV on a fixed proton target, predicted from our phenomenogical model (90% confidence intervals).Coupling α B = 1 is fixed.

FIG. 8 .
FIG. 8. Probability distributions of three kinematic quantities in B-boson electroproduction, electron scattering angle θ in the lab frame (left), outgoing electron energy E ′ relative to the beam energy E beam (center), and photon-proton center-of-mass energy W relative to the total electron-proton center-of-mass energy √ s tot (right), for E beam = 11 GeV (top) and 22 GeV (bottom).Different colored bands indicate predictions from our phenomenological model (90% confidence intervals) for different m B values, from m B = 0.5 GeV (lightest) to m B = 1.1 GeV (darkest).