Anomalous U(1) Gauge Bosons as Light Dark Matter in String Theory
Luis A. Anchordoqui, Ignatios Antoniadis, Karim Benakli, Dieter Lust
MMPP-2020-121LMU-ASC 34 / Anomalous U (1) Gauge Bosons as Light Dark Matter in String Theory
Luis A. Anchordoqui,
1, 2, 3
Ignatios Antoniadis,
4, 5
Karim Benakli, and Dieter L¨ust
6, 7 Department of Physics and Astronomy, Lehman College, City University of New York, NY 10468, USA Department of Physics, Graduate Center, City University of New York, NY 10016, USA Department of Astrophysics, American Museum of Natural History, NY 10024, USA Laboratoire de Physique Th´eorique et Hautes ´Energies - LPTHE Sorbonne Universit´e, CNRS, 4 Place Jussieu, 75005 Paris, France Albert Einstein Center, Institute for Theoretical Physics University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland Max–Planck–Institut f¨ur Physik, Werner–Heisenberg–Institut, 80805 M¨unchen, Germany Arnold Sommerfeld Center for Theoretical Physics Ludwig-Maximilians-Universit¨at M¨unchen, 80333 M¨unchen, Germany
Present experiments are sensitive to very weakly coupled extra gauge symmetries which motivates further in-vestigation of their appearance in string theory compactifications and subsequent properties. We consider exten-sions of the standard model based on open strings ending on D-branes, with gauge bosons due to strings attachedto stacks of D-branes and chiral matter due to strings stretching between intersecting D-branes. Assuming thatthe fundamental string mass scale saturates the current LHC limit and that the theory is weakly coupled, weshow that (anomalous) U (1) gauge bosons which propagate into the bulk are compelling light dark matter can-didates. We comment on the possible relevance of the U (1) gauge bosons, which are universal in intersectingD-brane models, to the observed 3 σ excess in XENON1T. The primary objective of the High Energy Physics (HEP)program is to find and understand what physics may lie be-yond the Standard SU (3) C ⊗ SU (2) L ⊗ U (1) Y Model (SM),as well as its connections to gravity and to the hidden sec-tor of particle dark matter (DM). This objective is pursued inseveral distinct ways. In this Letter, we explore one possiblepathway to join the vertices of the HEP triangle using stringcompactifications with large extra dimensions [1], where setsof D-branes lead to chiral gauge sectors close to the SM [2, 3].D-branes provide a nice and simple realization of non-abelian gauge symmetry in string theory. A stack of N identi-cal parallel D-branes eventually generates a U ( N ) theory withthe associated U ( N ) gauge group where the correspondinggauge bosons emerge as excitations of open strings ending onthe D-branes. Chiral matter is either due to strings stretchingbetween intersecting D-branes, or to appropriate projectionson strings in the same stack. Gravitational interactions aredescribed by closed strings that can propagate in all dimen-sions; these comprise parallel dimensions extended along theD-branes and transverse ones.String compactifications could leave characteristic foot-prints at particle colliders: • the emergence of Regge recurrences at parton collisionenergies √ s ∼ string mass scale ≡ M s = / √ α (cid:48) [4–6]; • the presence of one or more additional U (1) gauge sym-metries, beyond the U (1) Y of the SM [7–9].Herein we argue that the (anomalous) U (1) gauge bosonsthat do not partake in the hypercharge combination could be-come compelling dark matter candidates. Indeed, as notedelsewhere [10] these gauge fields could live in the bulk andthe four-dimensional U (1) gauge coupling would become in-finitesimally small in low string scale models, g ∼ M s / M Pl ,where M Pl is the Planck mass (for previous investigations indi ff erent regions of parameters and di ff erent string scenarios,see for example [11–13]). Note that for typical energies E ofthe order of the electron mass, the value of g is still bigger that the gravitational coupling ∼ E / M Pl , and the strength ofthe new force would be about 10 times stronger than grav-ity, where we have taken M s ∼ (cid:46) electronic recoils / keV (cid:46)
7, peaked around2.8 keV [15]. The total number of events recorded withinthis energy window is 285, whereas the expected backgroundis 232 ±
15. Taken at face value this corresponds to a sig-nificance of roughly 3 σ , but unknown backgrounds from tri-tium decay cannot be reliably ruled out [15]. Although theexcess is not statistical significant, it is tempting to imaginethat it corresponds to a real signal of new physics. A plethoraof models have already been proposed to explain the excess,in which the DM particle could be either the main compo-nent of the abundance in the solar neighborhood, n DM ∼ ( m DM / . − cm − , or else a sub-component of theDM population. Absorption of a ∼ . U (1) X gauge coupling to elec-trons of g X , e ff ∼ × − − × − [15–20]. For such smallmasses and couplings, the cosmological production should benon-thermal [17], avoiding constraints from structure forma-tion [21, 22]. Leaving aside attempts to fit the XENON1T ex-cess, we might consider a wider range of dark photon massesand couplings. For light and very weakly coupled dark pho-tons, the cooling of red giants and horizontal branch starsgive stronger or similar bounds on g X , e ff than direct detec-tion experiments [23, 24]. For instance, rescaling the bounds A point worth noting at this juncture, however, is that there are severalstellar systems that exhibit a mild preference for an over-e ffi cient coolingmechanism when compared to theoretical models [25]. Thus, the argument a r X i v : . [ h e p - t h ] J u l quoted in [19] leads to an upper bound g X , e ff (cid:38) − − − for m X varying from 10 to 100 keV. As an example, if wetake m X ∼
15 keV in agreement with the bound of ∼ g X , e ff (cid:46) × − . Obtainingsuch small values of masses and couplings for the dark photonare challenging as we will show.We start from ten-dimensional type I string theory com-pactified on a six-dimensional space of volume V M s . Therelation between the Planck mass, the string scale, the stringcoupling g s , and the total volume of the bulk V M s reads: M = g s M s V (2 π ) . (1)A hierarchy between the Planck and string scales can be dueto either a large volume V M s (cid:29) d the total number of dimen-sions that are large. For simplicity, we assume that they havea common radius R while the other 6 − d dimensions have aradius M − s . The U (1) X gauge fields live on a D(3 + δ X )-branethat wraps a δ X -cycle of volume V X , while its remaining fourdimensions extend into the uncompactified space-time. Thecorresponding gauge coupling is given by: g X = (2 π ) δ X + g s V X M δ X s . (2)Assuming all the δ X -cycles are sub-spaces of internal d largedimensions with the same radius, the substitution of (1) into(2) leads to: g X = π g s (cid:32) g s (cid:33) δ X / d (cid:32) M s M Pl (cid:33) δ X / d . (3)It is straightforward to see that to realize the weakest gaugeinteraction the volume seen by U (1) X must exhaust the totallarge internal volume suppressing the strength of gravitationalinteractions δ X = d (as in [16]), yielding g X = (cid:115) π g s M s M Pl ∼ × − (cid:32) . g s (cid:33) / (cid:18) M s
10 TeV (cid:19) , (4)where we have taken as reference values g s = . M s (cid:38)
10 TeV. The latter is a conservative bound from non-observation of stringy excitations at colliders [14] while aslightly stronger bound of order, but model dependent, canbe obtained from limits on dimension-six four-fermion opera-tors [26–29]. As for g s we will consider that it is in the range0 . − .
2, and could be fixed after a careful study of runningof the gauge couplings. In the case of toroidal compactifi-cations, the internal six-dimensional volume is expressed in can be turned around and the anomalous cooling could be interpreted asevidence for U (1) X production in dense star cores. terms of the parallel and transversal radii as V = (2 π ) d (cid:107) (cid:89) i = R (cid:107) i d ⊥ (cid:89) j = i R ⊥ j , (5)where now for each stack of Dp-branes we identify the corre-sponding d (cid:107) = δ . For instance, if the SM arise from D3-branesand the U (1) X from D7-branes with an internal space hav-ing four large dimensions all parallel to the D7-brane world-volume ( δ X = d = g X the result in (4).Another possibility for engineering extremely weak extragauge symmetries is to consider a scenario which allows verysmall value of g s . Such a possibility is provided by small in-stantons [30, 31] or Little String Theory (LST) [32, 33] wherewe localize the SM gauge group on Neuveu-Schwarz (NS)branes (dual to the D-branes).In the case of LST [32, 33], we start with a compactificationon a six-dimensional space of volume V with the Planck massgiven by (1) (up to a factor 2 in the absence of an orientifold).The internal space is taken as a product of a two-dimensionalspace, of volume V , times a four-dimensional compact space,of volume V . However, instead of D-brane discussed above,we assume that the SM degrees of freedom emerge on a stackof NS5-branes wrapping the two-cycle of volume V . We takefor simplicity this to be a torus made of two orthogonal circleswith radii R and R . The corresponding (tree-level) gaugecoupling is given by: g = R R (Type IIA) and g = R R M s (Type IIB);(6)thus, an order one SM coupling imposes R (cid:39) R (cid:39) M − s . Onthe other hand, the U (1) X is supposed to appear in the bulk andhas a coupling given by (2). If U (1) X arises from a D9-branethen: M = g s M s V V (2 π ) , and g X = (2 π ) g s V V M s . (7)Now, taking all the internal space radii to be of the order ofthe string length, M s V V (cid:39) (2 π ) , leads to: g X (cid:39) √ π (cid:114) M s M Pl ∼ × − (cid:18) M s
10 TeV (cid:19) / . (8)Note however that the U (1) X from a D-brane does not interactdirectly with the electrons of the SM on the NS5-brane. Suchinteraction could arise via a closed string exchange which islikely to be suppressed by two powers of the string coupling,leading to an e ff ective interaction of the order of 10 − .In heterotic strings compactified on K
3, of volume V K fibered over a two-dimensional base P of volume V P withintegrated volume < V K V P > , the Planck mass reads: M = π g s M s < V K V P > . (9)Taking the limit of instanton small size leads to emergenceof a gauge group, identified with the SM one, supported atparticular points on K g = π M s < V P > , (10)implying that to give phenomenologically acceptable values,the compactification radius should remain of order of thestring scale. The U (1) X is identified within the bulk theorydescending from the ten-dimensional gauge symmetry: g X = g s M s √ < V K V P > = √ π M s M Pl ∼ × − M s
10 TeV . (11)Taking < V K V P > (cid:39) < V K >< V P > , we see that the weak-ness of gravitational interactions, and a consequence of the U (1) X coupling, can be due either to a large volume of the K < V K > / ∼ GeV − or g s ∼ − for M s ∼
10 TeV.We turn now to the generation of a mass for the dark pho-ton. Let’s denote by v X the vacuum expectation value for theHiggs h X that breaks the U (1) X symmetry. The simplest quar-tic potential − µ X h X + λ X h X leads to v X = µ X / √ λ X , a Higgsmass of order µ X and a mass for the dark photon m X = g X µ X √ λ X = (cid:112) π g s (cid:32) g s (cid:33) δ/ d (cid:32) M s M Pl (cid:33) δ/ d v X . (12)This gives for d = δ = m X ∼ (cid:32) . g s (cid:33) / (cid:18) M s (cid:19) (cid:32) v X M s (cid:33) keV . (13)Taking v X (cid:39) M s , this leads to a mass of order 0 . . × eV when varying M s from 10 to 1000 TeV, and g s from 0 . .
02. For this region of the parameter space,the gauge coupling varies in the range 4 × − (cid:46) g X (cid:46) × − . Higher photon masses are of course easier to ob-tain with smaller number of d (cid:107) dimensions. For example, an M s ∼
10 TeV, and M s ∼
100 TeV lead respectively to m X ∼ g X ∼ × − , and m X ∼
270 keV, g X ∼ × − for δ X = d = g s ∼ . U (1) X becomes massive via a St¨uckelberg mechanism as a conse-quence of a Green-Schwarz (GS) anomaly cancellation [34,35], which is achieved through the coupling of twistedRamond-Ramond axions [36, 37]. The mass of the anoma-lous U (1) X can be unambiguously calculated by a direct one-loop string computation. Assuming the U (1) X arises from abrane wrapping δ X dimensions among the d large dimensions,it is given by m X = κ (cid:115) V a M s V X M δ s M s = κ (2 π ) δ x − √ g s M s M Pl δ X − δ ad M s , (14) Note that the U (1) is not necessarily anomalous in four dimensions. A masscan be generated for a non-anomalous U (1) by a six-dimensional (6d) GSterm associated to a 6d anomaly cancellation in a sector of the theory. where κ is the anomaly coe ffi cient (which is in general an or-dinary loop suppressed factor), V a is the two-dimensional in-ternal volume corresponding to the propagation of the axionfield [10] and δ a is the number of large dimensions in V a . For δ a =
0, it leads to: m X = κ (2 π ) δ x − √ g s M s M Pl δ X / d M s , (15)which gives ∼ . κ keV and ∼ κ keV and for M s ∼
10 TeVand M s ∼
100 TeV, respectively ( δ X = d = g s = . × − (cid:46) g X (cid:46) × − . The case δ a = δ X = d = m X = κ (2 π ) √ g s M s M Pl M s ∼ κ (cid:32) . g s (cid:33) (cid:18) M s
10 TeV (cid:19) keV . (16)For a concrete example of such case, consider 2 D7-branesintersecting in two common directions; namely, D : 1234and D : 1256, where 123456 denote the internal six direc-tions. Take now 1234 large and 56 small (order the stringscale) compact dimensions. The gauge fields of D have asuppression of their coupling by the 4-dimensional internalvolume V X while the states in the intersection of the two D7branes see only the 12 large dimensions and give 6 dimen-sional anomalies, cancelled by an axion living in the sameintersection, so V a is the volume of 12 only.We have seen that the tiny couplings are not trivial to obtainand lead often to too small dark photon masses. This issue canbe alleviated by resorting to the case where e ff ective smallercouplings of U (1) X to SM states are obtained when the darkphotons do not couple directly to the visible sector, but doit through kinetic mixing with ordinary photons. It can begenerated by non-renormalisable operators, but it is natural toassume 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 : (cid:15) γ X = eg X π (cid:88) i q ( i ) q ( i ) X ln m i µ ≡ eg X π C Log (17)where µ denotes the renormalization scale , where we ab-sorbed also the constant contribution. The e ff ective couplingto SM is then: g X , e ff = e (cid:15) γ X = α em g X π C Log ∼ × − g X C Log . (18)We can try to fit both desired values of g X , e ff and m X . For amass of the dark photon arising from a Higgs mechanism, wedetermine g x ∼ m X / M s , with v X ∼ M s , this constrains: C Log (cid:39) . × g X , e ff M s m X (cid:39) . (cid:18) g X , e ff × − (cid:19) (cid:18) M s
100 TeV (cid:19) (cid:18) m X . (cid:19) − . (19) In string theory, it is replaced by the string scale M s . A cancellation in the logarithm can be total, and the con-tribution appears at higher loops [38], or partial, for in-stance between particles with (order one) charges ( q ( i ) , q ( i ) X )and ( q ( j ) , q ( j ) X = − q ( i ) X ) and masses m j = m i + ∆ m i j . For ∆ m i j (cid:28) m i , we have an approximation: C Log ∼ (cid:88) i , j ∆ m i j m i . (20)We shall now discuss more explicitly the emergence of suchextra abelian gauge groups in D -brane models. The minimalembedding of the SM particle spectrum requires at least threebrane stacks [39] leading to three distinct models of the type U (3) ⊗ U (2) ⊗ U (1) that were classified in [39, 40]. Only one ofthem, model (C) of [40], has baryon number as a gauge sym-metry that guarantees proton stability (in perturbation theory),and can be used in the framework of low mass scale stringcompactifications. In addition, because the charge associatedto the U (1) of U (2) does not participate in the hyperchargecombination, U (2) can be replaced by the symplectic Sp (1)representation of Weinberg-Salam SU (2) L , leading to a modelwith one extra U (1) added to the hypercharge [41]. Note thatthe abelian factor associated to the U (2) stack of D-branescouples to the lepton doublet, and consequently this anoma-lous U (1) cannot be a good dark matter candidate, because theleft-handed neutrinos make it unstable. One can add to thesethree stacks another D9-brane which will provide the U (1) X which will mix with the photon through loops of states livingin the intersections of the D9 and the U (3) and U (1) stacks.The dark U (1) X is of course unstable as it decays to three or-dinary photons. However, the partial decay width is found tobe [42] Γ X → γ ∼ − (cid:18) m X . (cid:19) (cid:18) g X , e ff × − (cid:19) Gyr − , (21)and so for the range of small gauge coupling considered here,the life-time is big enough to allow it be a viable candidate fordark matter.Actually, the SM embedding in four D-brane stacks leads tomany more models that have been classified in [43, 44]. Thetotal gauge group of interest here, G = U (3) C ⊗ U (2) L ⊗ U (1) ⊗ U (1) X (22) = SU (3) C ⊗ U (1) C ⊗ SU (2) L ⊗ U (1) L ⊗ U (1) ⊗ U (1) X , contains four abelian factors. The non-abelian structure deter-mines the assignments of the SM particles. The quark doublet Q corresponds to an open string with one end on the colorstack of D-branes and the other on the weak stack. The anti-quarks u c and d c must have one of their ends attached to thecolor branes. The lepton doublet and possible Higgs doubletsmust have one end on the weak set of branes. Per contra, theabelian structure is not fixed because the U (1) Y boson, whichgauges the usual electroweak hypercharge symmetry, couldbe a linear combination of all four abelian factors. However,herein we restrict ourselves to models in which the bulk U (1) X TABLE I: Chiral fermion spectrum of the D-brane model (3) .Fields Representation q C q L q q X q Y Q (3 , u c ( ¯ , − − − d c ( ¯ , − − L (1 , − − e c (1 , (5) .Fields Representation q C q L q q X q Y Q (3 , − u c ( ¯ , − − d c ( ¯ , − L (1 , − − e c (1 , does not contribute to the hypercharge, in order to avoid an un-realistically small gauge coupling. Of particular interest hereare models (3) and (5) of reference [43]. The general proper-ties of their chiral spectra are summarized in Table I and II.One can check by inspection that for both models the hyper-charge, q Y = − q C + q L + q for model (3)q Y = q C + q L + q for model (5) (23)is anomaly free. In addition, the U (1) X is long-lived (becauseit only couples to the e c and to either u c or d c ) and therefore aviable DM candidate.In summary, we have investigated a model of light darkmatter based on ubiquitous U (1) gauge bosons of D-branestring compactifications. We have shown that this modelcan accommodate the excess of events with 3 σ significanceover background recently observed at XENON1T. The modelis fully predictive, and can be confronted with future datafrom dark matter direct-detection experiments, LHC Run 3searches, and astrophysical observations.The work of L.A.A. is supported by the U.S. NationalScience Foundation (NSF Grant PHY-1620661) and the Na-tional Aeronautics and Space Administration (NASA Grant80NSSC18K0464). The research of I.A. is funded in partby the “Institute Lagrange de Paris”, and in part by a CNRSPICS grant. The work of K.B. is supported by the Agence Na-tionale de Recherche under grant ANR-15-CE31-0002 “Hig-gsAutomator”. The work of D.L. is supported by the OriginsExcellence Cluster. Any opinions, findings, and conclusionsor recommendations expressed in this material are those of theauthors and do not necessarily reflect the views of the NSF orNASA. [1] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos andG. R. Dvali, New dimensions at a millimeter to a Fermiand superstrings at a TeV, Phys. Lett. B , 257-263 (1998) doi:10.1016 / S0370-2693(98)00860-0 [arXiv:hep-ph / , 71 (2005) doi:10.1146 / annurev.nucl.55.090704.151541[hep-th / , 1 (2007)doi:10.1016 / j.physrep.2007.04.003 [hep-th / , 171603 (2008) doi:10.1103 / PhysRevLett.100.171603[arXiv:0712.0386 [hep-ph]].[5] L. A. Anchordoqui, H. Goldberg, D. L¨ust, S. Nawata,S. Stieberger and T. R. Taylor, Dijet signals for low massstrings at the LHC, Phys. Rev. Lett. , 241803 (2008)doi:10.1103 / PhysRevLett.101.241803 [arXiv:0808.0497 [hep-ph]].[6] L. A. Anchordoqui, I. Antoniadis, D. C. Dai, W. Z. Feng,H. Goldberg, X. Huang, D. L¨ust, D. Stojkovic and T. R. Tay-lor, String resonances at hadron colliders, Phys. Rev. D , no.6, 066013 (2014) doi:10.1103 / PhysRevD.90.066013[arXiv:1407.8120 [hep-ph]].[7] D. M. Ghilencea, L. E. Ibanez, N. Irges and F. Quevedo,TeV scale Z -prime bosons from D-branes, JHEP ,016 (2002) doi:10.1088 / / / /
016 [arXiv:hep-ph / Z (cid:48) -gauge bosons as harbingersof low mass strings, Phys. Rev. D , 086003 (2012)doi:10.1103 / PhysRevD.85.086003 [arXiv:1107.4309 [hep-ph]].[9] M. Cvetic, J. Halverson and P. Langacker, Implications of stringconstraints for exotic matter and Z (cid:48) s beyond the standardmodel, JHEP , 058 (2011) doi:10.1007 / JHEP11(2011)058[arXiv:1108.5187 [hep-ph]].[10] I. Antoniadis, E. Kiritsis and J. Rizos, Anomalous U(1)sin type 1 superstring vacua, Nucl. Phys. B , 92-118 (2002) doi:10.1016 / S0550-3213(02)00458-3 [arXiv:hep-th / , 124 (2008)doi:10.1088 / / / /
124 [arXiv:0803.1449 [hep-ph]].[12] C. P. Burgess, J. P. Conlon, L. Y. Hung, C. H. Kom,A. Maharana and F. Quevedo, Continuous Global Symme-tries and Hyperweak Interactions in String Compactifications,JHEP , 073 (2008) doi:10.1088 / / / / , 027 (2009) doi:10.1088 / / / /
027 [arXiv:0909.0515 [hep-ph]].[14] A. M. Sirunyan et al. [CMS], Search for high mass dijet res-onances with a new background prediction method in proton-proton collisions at √ s =
13 TeV, JHEP , 033 (2020) doi:10.1007 / JHEP05(2020)033 [arXiv:1911.03947 [hep-ex]].[15] E. Aprile et al. [XENON], Observation of excess electronic re-coil events in XENON1T, [arXiv:2006.09721 [hep-ex]].[16] K. Benakli, C. Branchina and G. La ff orgue-Marmet, U (1) mix-ing and the weak gravity conjecture, [arXiv:2007.02655 [hep-ph]].[17] G. Alonso-Alvarez, F. Ertas, J. Jaeckel, F. Kahlhoefer andL. J. Thormaehlen, Hidden photon dark matter in the light ofXENON1T and stellar cooling, [arXiv:2006.11243 [hep-ph]].[18] G. Choi, M. Suzuki and T. T. Yanagida, XENON1Tanomaly and its implication for decaying warm dark matter,[arXiv:2006.12348 [hep-ph]].[19] H. An, M. Pospelov, J. Pradler and A. Ritz, New limits ondark photons from solar emission and keV scale dark matter,[arXiv:2006.13929 [hep-ph]].[20] N. Okada, S. Okada, D. Raut and Q. Shafi, Dark matter Z (cid:48) and XENON1T excess from U (1) X extended standard model,[arXiv:2007.02898 [hep-ph]].[21] D. Gilman, S. Birrer, A. Nierenberg, T. Treu, X. Du andA. Benson, Warm dark matter chills out: constraints on thehalo mass function and the free-streaming length of dark mat-ter with eight quadruple-image strong gravitational lenses,Mon. Not. Roy. Astron. Soc. , no.4, 6077-6101 (2020)doi:10.1093 / mnras / stz3480 [arXiv:1908.06983 [astro-ph.CO]].[22] N. Banik, J. Bovy, G. Bertone, D. Erkal and T. J. L. de Boer,Novel constraints on the particle nature of dark matter from stel-lar streams, [arXiv:1911.02663 [astro-ph.GA]].[23] H. An, M. Pospelov and J. Pradler, New stellar con-straints on dark photons, Phys. Lett. B , 190-195(2013) doi:10.1016 / j.physletb.2013.07.008 [arXiv:1302.3884[hep-ph]].[24] M. Fabbrichesi, E. Gabrielli and G. Lanfranchi, The dark pho-ton, [arXiv:2005.01515 [hep-ph]].[25] M. Giannotti, I. Irastorza, J. Redondo and A. Ringwald, CoolWISPs for stellar cooling excesses, JCAP , 057 (2016)doi:10.1088 / / / /
057 [arXiv:1512.08108[astro-ph.HE]].[26] E. Accomando, I. Antoniadis and K. Benakli, Looking forTeV scale strings and extra dimensions, Nucl. Phys. B ,3-16 (2000) doi:10.1016 / S0550-3213(00)00123-1 [arXiv:hep-ph / ,055012 (2000) doi:10.1103 / PhysRevD.62.055012 [arXiv:hep-ph / , 044 (2001) doi:10.1088 / / / /
044 [arXiv:hep-th / , 377-393 (2003) doi:10.1016 / S0550-3213(03)00404-8[arXiv:hep-ph / , 541-559 (1996) doi:10.1016 / / , 83-88 (2000) doi:10.1016 / S0370-2693(99)01422-7 [arXiv:hep-th / , 055 (2001) doi:10.1088 / / / /
055 [arXiv:hep-th / , 081602 (2012) doi:10.1103 / PhysRevLett.108.081602 [arXiv:1102.4043[hep-ph]].[34] M. B. Green and J. H. Schwarz, Anomaly cancellation in super-symmetric D =
10 gauge theory and superstring theory, Phys.Lett. B , 117-122 (1984) doi:10.1016 / ff ects on the string world sheet (II), Nucl. Phys. B , 319-363 (1987) doi:10.1016 / (1992) 196doi:10.1016 / / U (1)’sin type I and type IIB D = N = , 112-138 (1999) doi:10.1016 / S0550-3213(98)00791-3[arXiv:hep-th / (2019) no.9, 095001 doi:10.1103 / PhysRevD.100.095001[arXiv:1909.00696 [hep-ph]].[39] I. Antoniadis, E. Kiritsis and T. N. Tomaras, A D-brane alternative to unification, Phys. Lett. B , 186-193 (2000) doi:10.1016 / S0370-2693(00)00733-4 [arXiv:hep- ph / , 120-140 (2005)doi:10.1016 / j.nuclphysb.2005.03.005 [arXiv:hep-th / , 095009 (2007)doi:10.1103 / PhysRevD.75.095009 [arXiv:hep-th / , 005 (2009) doi:10.1088 / / / /
005 [arXiv:0811.0326 [hep-ph]].[43] I. Antoniadis, E. Kiritsis, J. Rizos and T. N. Tomaras, D-branes and the standard model, Nucl. Phys. B , 81-115 (2003) doi:10.1016 / S0550-3213(03)00256-6 [arXiv:hep-th / , 83-146 (2006)doi:10.1016 / j.nuclphysb.2006.10.013 [arXiv:hep-th //