Anomalous U(1) Gauge Bosons and String Physics at the Forward Physics Facility

We show that experiments at the Forward Physics Facility, planned to operate near the ATLAS interaction point during the LHC high-luminosity era, will be able to probe predictions of Little String Theory by searching for anomalous U(1) gauge bosons living in the bulk. The interaction of the abelian broken gauge symmetry with the Standard Model is generated at the one-loop level through kinetic mixing with the photon. Gauge invariant generation of mass for the U(1) gauge boson proceeds via the Higgs mechanism in spontaneous symmetry breaking, or else through anomaly-cancellation appealing to Stueckelberg-mass terms. We demonstrate that FASER2 will be able to probe string scales over roughly two orders of magnitude: 10^5

We show that experiments at the Forward Physics Facility, planned to operate near the ATLAS interaction point during the LHC high-luminosity era, will be able to probe predictions of Little String Theory by searching for anomalous U(1) gauge bosons living in the bulk. The interaction of the abelian broken gauge symmetry with the Standard Model is generated at the one-loop level through kinetic mixing with the photon. Gauge invariant generation of mass for the U(1) gauge boson proceeds via the Higgs mechanism in spontaneous symmetry breaking, or else through anomaly-cancellation appealing to Stückelberg-mass terms. We demonstrate that FASER2 will be able to probe string scales over roughly two orders of magnitude: 10 5 M s /TeV 10 7 .
In Ref. [1] we investigated the sensitivity of dark matter direct detection experiments to extremely weakly coupled extra U(1) gauge symmetries which are ubiquitous in D-brane string compactifications [2,3]. In this follow up to the dark matter work we particularize our investigation to experiments planned to operate at the HL-LHC Forward Physics Facility (FPF) [4,5]. Before proceeding, we pause to stress that our investigation will be framed within the context of Little String Theory (LST), which allows us to take the string coupling g s of arbitrary small values [6,7]. This contrasts with previous literature on hidden U(1) in string theory, which pivots on the volume of the internal space rather than on g s . We focus attention on the FPF's second generation For-wArd Search ExpeRiment (FASER2). 1 FASER2 will be shielded from the ATLAS interaction point by 200 m of concrete and rock, creating an extremely low-background environment for searches of long-lived particles traveling unscathed along the beam collision axis. Herein, we are interested in searches for light, very weakly-interacting vector fields that couple through kinetic mixing to the hypercharge gauge boson or, at low energies, effectively to the Standard Model photon (SM). At hadron colliders like the LHC, dark U(1) X gauge bosons of mass m X can be abundantly produced through proton bremsstrahlung or via the decay of heavy mesons. Indeed, over the lifetime of the HL-LHC there will be 4×10 17 neutral pions, 6×10 16 η mesons, 2×10 15 D mesons, and 10 13 B mesons produced in the direction of FASER2. The U(1) X discovery potential of FASER2 in the (m X , g X,eff ) plane is shown in Fig. 1, where we have defined the effective kinetic mixing parameter g X,eff ≡ e γX , and where e is the elementary charge and γX is the physical kinetic mixing parameter. We note in passing that complementary measurements 1 FASER has been already installed in the LHC tunnel and will collect data during Run 3 [8].
We now turn to demonstrate that LST provides a compelling framework for engineering very weak extra gauge symmetries with masses 500 < m X /MeV < 800 and 10 −8 < g X,eff < 10 −7 . The SM gauge group is localized on Neuveu-Schwarz (NS) branes (dual to the D-branes). However, the U(1) X gauge field could live in the bulk and if so its four-dimensional gauge coupling becomes infinitesimally small [17].
We consider a compactification on a six-dimensional space with the Planck mass given by 6 (1) (up to a factor 2 in the absence of an orientifold), where M s is the string scale and we have taken the internal space to be a product of a two-dimensional space, of volume V 2 , times a four-dimensional compact space, of volume V 4 . We further consider that the SM degrees of freedom emerge on a stack of NS5-branes wrapping the two-cycle of volume V 2 . For simplicity, we assume that the internal space is characterized by a torus made of two orthogonal circles with radii R 1 and R 2 . The corresponding (tree-level) gauge coupling is given by: (2) hence, an order one SM coupling imposes R 1 R 2 M −1 s . The U(1) X gauge field lives on a D(3 + δ X )-brane that wraps a δ X -cycle of volume V X , while its remaining four dimensions extend into the uncompactified space-time. The corresponding gauge coupling is given by, If U(1) X arises from a D7-brane (3) can be recast as We further assume that all the internal space radii are of the order of the string length, M 6 s V 2 V 4 (2π) 6 , so that (1) and (4) yield, The dark gauge boson acquires a mass through the Higgs mechanism, with v X the vacuum expectation value for the Higgs h X that breaks the U(1) X symmetry. We consider the simplest quartic potential −µ 2 X h 2 X + λ X h 4 X , which gives v X = µ X / √ 2λ X , a Higgs mass of order µ X , and a mass for the dark gauge boson where we have taken λ X to be O(1), and where d is the total number of dimensions that are large. Throughout our calculations we take d = δ X . Alternatively, the abelian gauge field U(1) X could acquire a mass via a Stückelberg mechanism as a consequence of a Green-Schwarz (GS) anomaly cancellation [18,19], which is achieved through the coupling of twisted Ramond-Ramond axions [20,21]. The mass of the anomalous U(1) X can be unambiguously calculated through a direct one-loop string computation, and is given by where κ is the GS anomaly coefficient (which is a numerical factor of order 10 −1 to 10 −2 times √ g s ∝ g X ), V a is the two-dimensional internal volume corresponding to the propagation of the axion field [17] and δ a is the number of large dimensions in V a . 2 To develop some sense for the orders of magnitude involved, in our calculations we take δ a = 2 and δ X = d = 4. With this in mind, (7) can be rewritten as For a concrete example of this set up, we envision 2 D7-branes intersecting in two common directions; viz., D7 1 : 1234 and D7 2 : 1256, where 123456 denote the internal six directions. Next, we take 1234 to be large and 56 to be small (i.e, order the string scale) compact dimensions. The gauge fields of D7 1 have a suppression of their coupling by the 4-dimensional internal volume V X while the states in the intersection of the two D7 branes only see the 12 large dimensions and lead to 6 dimensional anomalies, which are cancelled by an axion living in the same intersection, so V a is the volume of 12 only. As noted above, the U(1) X does not couple directly to the visible sector, but does it via kinetic mixing with ordinary photons. This coupling can be generated by nonrenormalizable operators, but it is natural to assume that it is generated by loops of states carrying charges (q (i) , q (i) X ) under the two U(1)'s and having masses m i : where µ 2 denotes the renormalization scale (which in string theory is replaced by M s ), and where we absorbed also the constant contribution. The effective coupling to SM is then given by Using (5) and (6), as well as (8) or (10) we scan over the LST parameter space. Our results are encapsulated in Fig. 1, where we show representative values of the (m X , g X,eff ) plane, with LST model parameters listed in Table I. In our calculations we set C Log ∼ 3 [22]. A point worth noting at this juncture is that for g X,eff 5 × 10 −8 , the associated abelian gauge boson can only acquire a mass via the Higgs mechanism, since the required GS numerical factor becomes unnaturally small. We can explicitly see in the figure that the LST parameter space region probable by FASER2 spans roughly two orders of magnitude in the string scale.
In summary, we have shown that FASER2 will be able to probe a region of the LST parameter space by searching for abelian gauge bosons living in the bulk. From (6) we see that the effective coupling g X,eff depends on the product √ M s × C Log . Similar O(1) logarithmic terms appear when computing threshold corrections to gauge couplings. For C Log of order one, we can see in Fig. 1 that there is a minimum value of the string scale, roughly of order 10 5 TeV, which is in the FASER2 probable region. There is also a maximum value that can be tested in this region, M s 10 7 TeV. For the mass of the hidden Higgs, one needs some hierarchy given by v X /M s as shown in Table I. Given these considerations, the mass of the hidden U(1) X scales as (v x /M s ) × M 3/2 s , so the hierarchy increases as M 3/2 s for fixed m X . In closing, we note that the SM singlet scalar field S would couple to the SM Higgs doublet H, yielding a portal into the dark sector [23]. However, for µ X m H , the S H mixing angle 1, where m H is the Higss mass [24]. Using the values of v X given in Table I, it is straightforward to see that the Higgs portal does not generate additional bounds on the model and that the hidden scalar is out of the LHC reach; e.g., for M s ∼ 8 × 10 4 TeV, we have µ X ∼ 100 TeV.
The work of L.A.A. is supported by the U.S. National Science Foundation (NSF Grant PHY-2112527). The work of K.B. is supported by the Agence Nationale de Recherche under grant ANR-15-CE31-0002 "HiggsAutomator". The work of D.L. is supported by the Origins Excellence Cluster.