Scrutinizing a massless dark photon:
basis independence and new observables
Abstract
A new gauge boson field can have renormalizable kinetic mixing with the standard model (SM) gauge boson field . This mixing induces interactions of with SM particles even though starts as a dark photon without such interactions. If the is not broken, both the dark photon field and the photon field are massless. One cannot determine which one of them is the physical dark photon or the photon by just looking at kinetic terms in the Lagrangian. We further show that the kinetic mixing does not leave traces in electromagnetic and weak interactions if only SM particles are involved. The kinetic mixing provides a portal for probing the dark sector beyond the SM. Missing energy due to the escape of dark sector particles in probes that are sensitive only to SM particles leads to observable effects of the kinetic mixing massless dark photon. We study examples for a Higgs boson decays into a monophoton plus missing energy, and a boson decays into missing energy.
pacs:
PACS numbers:I Introduction
A new gauge symmetry with a gauge boson field can mix with the gauge boson field of in the standard model (SM) through a renormalizable kinetic mixing operator formed by the field strengths, , with Holdom (1986); Foot and He (1991); Okun (1982); Galison and Manohar (1984). If the SM particles are all uncharged under the , it is expected to have no interaction with SM particles. In this case is dubbed as a dark photon field. However, the kinetic mixing term can induce interactions between and the SM particles. This has many interesting consequences in low energy and high energy phenomena from particle physics, astrophysics to cosmology perspectives. Dark photon has been searched for in a number of different contexts experimentally Essig et al. (2013); Alexander et al. (2016).
If the dark photon field receives a finite mass, one can easily identify the physical dark photon and photon after the fields are redefined to have the canonical form for the gauge bosons, in which the kinetic terms are diagonal. However, for the case that the dark photon is trivially massless, the situation is different. If one just looks at the kinetic terms of and , the canonical form is invariant under any orthogonal transformation, then one cannot tell any difference before and after the transformation. Therefore, which combination of and in the canonical form corresponds to the physical photon or dark photon cannot be determined del Aguila et al. (1995).
Phenomenology of a massless dark photon has drawn a vast of attention Dobrescu (2005); Fabbrichesi et al. (2018, 2017); Gabrielli et al. (2014); Biswas et al. (2016); Daido et al. (2018); Huang and Lee (2018); Ackerman et al. (2009). One needs to be clear about how the massless dark photon interacts with SM particles to have correct interpretations of the results. The interactions of photon, boson and dark photon fields to the SM currents must be consistently defined to pin down the massless dark photon itself. We find that two commonly used ways to remove the mixing term are actually related through an orthogonal transformation. But the angle that describes the general orthogonal transformation does not affect how the massless dark photon and photon interact with SM particles. We show that effects of the kinetic mixing does not leave traces in the electromagnetic (EM) and weak interactions involving only SM particles, such as of a charged lepton and in the processes Higgs decays into two photons, and boson decays into SM particles. To detect massless dark photon effects, information about dark current needs to be known in some way. Missing energy due to the escape of dark sector particles in probes that are sensitive only to SM particles leads to observable effects of the kinetic mixing massless dark photon. We give some possible ways to detect the massless dark photon effects.
Ii Eliminating kinetic mixing for a massless dark photon
With the kinetic mixing, the kinetic terms of and and their interactions with other particles can be written as
(1) 
Here and denote interaction currents of gauge fields and , respectively.
To write the above Lagrangian in the canonical form one needs to diagonalize the kinetic terms of and . Let us consider two commonly used ways of removing the mixing, namely, a) Holdom (1986); Dobrescu (2005) the mixing term is removed in such a way that dark photon in the canonical form does not couple to hypercharge current ^{5}^{5}5In the literature case a) is widely used not only for a massless dark photon but also for a very light one Ahlers et al. (2007); Redondo (2008); Soper et al. (2014); Danilov et al. (2018), which is sometimes called “paraphoton” Okun (1982); Holdom (1986)., and b) Foot and He (1991); Babu et al. (1998) the hypercharge field in the canonical form does not couple to dark current produced by some dark particles with charges, which is widely used in the studies of a massive dark photon or Foot and He (1991); He et al. (2017, 2018); Babu et al. (1998); Curtin et al. (2015). For the cases a) and b), making the Lagrangian in the canonical form will be
(2)  
After electroweak symmetry breaking (EWSB), the hypercharge field and the neutral component of the gauge field can be written in the combinations of the ordinary photon field and the field as follows
(3) 
where and with being the weak mixing angle. Meanwhile, the field receives a mass .
The general Lagrangian that describes , and fields kinetic energy, and their interactions with the electromagnetic (EM) current , neutral boson current and dark current are given by
(4)  
where the boson mass term is included.
The dark photon may be also massive. There are two popular ways of generating dark photon mass giving rise to different phenomenology. One of them is the “Higgs mechanism”, in which the is broken by the vacuum expectation value (vev) of a SM singlet, which is charged under . In this case, the mixing of Higgs doublet and the Higgs singlet offers the possibility of searching for dark photon at colliders in Higgs decays Curtin et al. (2015). The other is the “Stueckelberg mechanism” Kors and Nath (2004); Stueckelberg (1938) in which an axionic scalar was introduced to allow a mass for without breaking . An interesting application of this mechanism to a gauged BL symmetry has been discussed in Ref. Heeck (2014). In our later discussion our concern is whether the dark photon has a mass or not, and therefore we only need to discussion the effect of a mass term in the above equation. The fields and their mass, due to electroweak symmetry breaking of the SM, are not affected.
The requirements for cases a) and b) can be equivalently expressed as no dark photon interaction with and no photon interaction with , respectively. These two cases can be achieved by defining
to obtain the Lagrangian in the case of ,
We clearly see that the properties for case a) and case b) are explicit. In both cases the boson mass is shifted as with . Note that in the above two ways of removing the kinetic mixing term, the boson interactions are the same in form.
The dark photon fields in the above are and , respectively. It has been argued using Eq. (II) that dark photon does not interact with SM particles at the treelevel Dobrescu (2005); Fabbrichesi et al. (2018, 2017); Gabrielli et al. (2014). But if one uses Eq. (II), the dark photon does interact with SM particles at the treelevel. The statements are in conflict with each other. This conflict lies in the definition for a dark photon.
If one just looks at the first two kinetic terms in Eqs. (II) (II), they are the same in form and invariant under an orthogonal transformation of and , or and . In fact, there are related by
(8) 
But for the case with , the situation is different. One can completely determine the physical states among , and . With , we need to add a mass term to the Lagrangian. In cases a) and b), they have the following forms
(9) 
To identify the physical photon, we find that the fields defined in case b) is more convenient to use since the field is already the physical massless photon field without further mass diangonalization. To obtain physical and , in case b), one needs to diagonalize the mass matrix in basis,
(10) 
to obtain the mass eigenstates
(11) 
with
(12) 
The interactions of physical photon, boson and dark photon can be determined accordingly without ambiguities. Expressing , and in terms of , and , one also obtain physical gauge boson interactions with SM and dark sector particles. A consistent treatment for case a) will lead to the same final results.
Let us come back to the situation with and discuss whether one can determine what the physical photon and massless dark photon are. To this end we use a most general basis based on case b)
(13) 
where and . For , and as compared with Eq. (8). For spanning from 0 to , all possible ways of removing the kinetic mixing to have a canonical form of , and fields can be covered. We have the following Lagrangian for the most general form for interactions for , and
(14)  
Note that in the above and are not what to be identified as physical photon and dark photon. The physical photon and dark photon should be the fields which respond to and to produce signal, that is, the components in and to and , respectively. Experimentally, the signals responding to cannot be detected by laboratory probes and becomes missing energy . In next section, we will use and to stand for the fields and for convenience.
Iii Physical effects of a massless dark photon
Let us first study how EM interaction is affected by the kinetic mixing of a massless dark photon. A wellmotivated observable is the anomalous magnetic dipole moment of fermion. There is a longstanding discrepancy between the experimental value and the SM prediction of the anomalous magnetic moment of the muon, M. Tanabashi et al. (Particle Data Group) (2018) . A lot of theoretical efforts have been made to explain this anomaly, see Refs. Baek et al. (2001); Pospelov (2009) for the “solutions” with a dark photon.
The oneloop diagram that contributes to is shown in Fig. 1. This part of contribution for a massless dark photon field can be easily obtained by rescaling the oneloop EM correction by a factor , that is
(15) 
where . Explaining with seemingly indicates physical effect of the massless dark photon depending on the artificial rotation angle .
However, since both photon and dark photon fields are massless, their contributions to muon should be included consistently. Apart from the massless dark photon field , the photon field in the loop should also be considered. Similar to , we obtain
(16) 
The sum of these two contributions is therefore
(17) 
where with the redefinition of the electric charge . This amounts to redefine . The effect of kinetic mixing term can therefore to be absorbed into redefinition of the electric charge that is independent of the angle . It is interesting to note that there is also a nonzero muon couple to (see Fig. 1), under the influence of a nonzero .
In the above we have seen that the observable effects of EM interaction is independent of in the example of muon and no beyond SM effects show up. From Eq. (14), the boson interaction is already independent of . One can ask whether any physical effects induced by the kinetic mixing show up in the weak interaction involving only SM particles.
In the minimal massless dark model we considered, there is no modification of boson interactions. Thus no new effects will show up in weak interactions involving the bosons. The mass of , the charged current and their couplings to SM fermions are not affected by the field redefinition as discussed in section II, we have , and the charged current . But the boson mass is modified as . Therefore, the onshell definition of weak mixing angle is modified accordingly as Sirlin (1980); Hollik (1990); Bardin et al. (1997) with . We have the following redefined currents
(18) 
where and indicate the upper and lower components of the left handed fermion doublets, respectively. is the gauge coupling, . The axialvector and vector couplings of boson are given by
(19) 
with . Here indicates SM fermion which has the isospin to be and for the upper and down doublet components.
We thus obtain parameter
(20) 
which is defined as the ratio of low energy neutral current to charged current scattering amplitudes M. Tanabashi et al. (Particle Data Group) (2018); Langacker (2010); Barger and Phillips (1987).
Naively, since is modified to be , this change seems should generate a nonzero value for the parameter and therefore . However, in our case, the neutral current also gets modified by a factor of . When taking the ratio for , the factor cancels out.
We conclude that if processes involve only SM particles and only EM and weak interactions are probed, there is no physical effect showing up due to a nonzero if dark photon is exactly massless.
Where can the kinetic mixing effect then be detected? Interaction with dark current must be involved in order to see any physical effect. Since dark current cannot be detected using detectors that are sensitive only to SM particles, the effects will be in the form of missing energy , including onshell or offshell dark photon. We consider processes involving two massless gauge bosons in the final states from the Higgs boson decay or protonproton collision as shown in Fig. 2. To obtain the final results, one starts from and analyze how and produce signal in the detectors. If or is detected by , it is identified as a photon . If or is detected by , it is identified as dark photon . Thus the signature for the diagram in Fig. 2 can be

: diphoton ;

: monophoton ;

: .
In case of , the collider signature is diphoton. The amplitude of including the contributions from and is proportional to
(21) 
The total effect amounts to the redefinition , just as in case, which is unobserved and independent of the angle . Therefore, the diphoton rate is equal to its SM value.
In case of , the signature is monophoton. It was argued that since does not couple to the EM currents in basis a), the monophoton signature in the minimal massless dark photon model can only arises from high dimensional operators Dobrescu (2005); Gabrielli et al. (2014). We however find that this signature does not vanish at tree level in the presence of dark current. To show how the physical effect arises, we calculate the monophoton rate in cases a) and b) in section II. In case b), can couple to the EM currents and dark current, while only interacts with the EM currents. The monophoton rate of diagrams with coupled to and coupled to is times the diphoton rate. On the other hand, in case a), although decouples from the SM, can interact with the EM currents and dark current. The diagrams with only must be considered in the monophoton process The monophoton rate is thus equal to that in case b). We also calculated the same quantity in the general basis in Eq. (14) and obtained the same independent of . The signature for the case of is missing energy. The event rate if times the diphoton rate.
Since there is no strong constraint on the kinetic mixing parameter for a massless dark photon Vogel and Redondo (2014), the above ratios can be sizable. Future data on monophoton and missing energy from a Higgs decays and collider processes can provide information and constraint on the parameter .
Finally, we would like to point out that the physical effect may also show up in boson decays into dark sector particles through the interaction in Eq. (14). Such decays will increase the invisible decay width of boson if the dark sector particles are light enough. In the SM the invisible width of boson is caused by the decays into neutrino pairs. The combined data give the effective neutrino number to be M. Tanabashi et al. (Particle Data Group) (2018) . There is still space for invisible decay width caused by the decays into dark sector.
Iv Conclusions
In this work, we show that for the SM extended with a gauge field having kinetic mixing with SM gauge field, the physical massless dark photon cannot be distinguished from the photon if rewriting gauge fields in the canonical form is the only requirement for removing the kinetic mixing term in the case with . To make the points, we first show the details of two commonly used ways and show that they are related by an orthogonal transformation. Furthermore, one can arrive at a general mass eigenstate of photon and dark photon from case b) by an orthogonal transformation described by a rotation angle. We have shown that such a mixing does not leave traces in the EM and weak interactions if only SM particles are involved. When missing energy due to the escape of dark sector particles with the massless dark photon portal is measured, one can detect physical effects of the kinetic mixing massless dark photon. We study it in the Higgs decays into monophoton and missing energy, boson decays into dark sector particles and missing energy in collisions. We encourage experimental colleagues to carry out related experiments.
Acknowledgements.
We would like to thank YiLei Tang and Fang Ye for fruitful discussions. This work was supported in part by the NSFC (Grant Nos. 11575115 and 11735010), by Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education, and Shanghai Key Laboratory for Particle Physics and Cosmology (Grant No. 15DZ2272100), and in part by the MOST (Grant No. MOST 1062112M002003MY3 ).References
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