###### Abstract

In Universal Extra Dimension models, the lightest Kaluza-Klein (KK) particle is generically the first KK excitation of the photon and can be stable, serving as particle dark matter. We calculate the thermal relic abundance of the KK photon for a general mass spectrum of KK excitations including full coannihilation effects with all (level one) KK excitations. We find that including coannihilation can significantly change the relic abundance when the coannihilating particles are within about 20% of the mass of the KK photon. Matching the relic abundance with cosmological data, we find the mass range of the KK photon is much wider than previously found, up to about 2 TeV if the masses of the strongly interacting level one KK particles are within five percent of the mass of the KK photon. We also find cases where several coannihilation channels compete (constructively and destructively) with one another. The lower bound on the KK photon mass, about 540 GeV when just right-handed KK leptons coannihilate with the KK photon, relaxes upward by several hundred GeV when coannihilation with electroweak KK gauge bosons of the same mass is included.

hep-ph/0509118

The Abundance of Kaluza-Klein Dark Matter

[5mm] with Coannihilation

Fiona Burnell and Graham D. Kribs

[5mm]

Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544

[1mm]

Department of Physics and Institute of Theoretical Science,

University of Oregon, Eugene, OR 97403

## 1 Introduction

One of the most important astrophysical challenges is to understand the nature and identity of dark matter. Perhaps the most interesting candidate is a neutral, stable, weakly interacting particle arising from physics beyond the Standard Model. This is consistent with a wide range of data, such as measurements of the cosmic microwave background (CMB) radiation that imply the abundance of dark matter is roughly a factor of 6 times the abundance of baryons [1], as well as comparisons between large scale structure simulations and galaxy survey observations that suggest the bulk of dark matter is cold [2].

Universal Extra Dimension (UED) models [3] (see also [4]), in which all of the Standard Model fields propagate in extra dimensions, provide an interesting example of Kaluza-Klein (KK) dark matter [5, 6, 7]. This is because bulk interactions do not violate higher dimensional momentum conservation (KK number), and in these models all of the couplings among the Standard Model particles arise from bulk interactions. To generate 4D chiral fermions, the extra compact dimension(s) must be modded out by an orbifold. For five dimensions this is , while in six dimensions is suitable and has other interesting properties [3] including motivation for three generations from anomaly cancellation [8] and the prevention of fast proton decay [9]. An orbifold does, however, lead to some of the less appealing aspects of the model. Brane-localized terms can be added to both orbifold fixed points that violate KK number. If these brane localized terms are symmetric under the exchange of the two orbifold fixed points, then a remnant of KK number conservation remains, called KK parity. All odd-level KK modes are charged under this discrete symmetry thereby ensuring that the lightest level-one KK particle (LKP) does not decay. The stability of the LKP suggests it could well be an interesting dark matter candidate.

We calculate the thermal relic abundance of dark matter including coannihilation with all level-one KK particles. This is an important calculation since there are many level-one KK particles that are near enough in mass to the LKP that they are expected to be important to determine the relic abundance. The identity of the LKP depends on the mass spectrum of the first KK level. We assume it is the lightest KK photon. This has been widely advocated for both its properties as dark matter and is favored by the contributions to the masses of the level one KK particles from one-loop radiative corrections [5, 7]. The spectrum of level-one KK particle masses is also not known. However, certain finite and log-enhanced one-loop radiative corrections [10] to the masses have been calculated in [5]. The size of these corrections depend on two unknowns: the size of the log, i.e., the cutoff scale of the theory, and the size of the matching corrections evaluated at the cutoff scale. Generically one would expect the calculable log-enhanced corrections to dominate over the uncalculable matching corrections. However, the log is not particularly large relative to the finite matching corrections. Indeed, should the cutoff scale be close to the scale of the KK particle masses, uncalculable matching corrections would be of order the calculable log-enhanced corrections, and then one would like to know the relic abundance for a general spectrum. This is the calculation we carry out in this paper.

We present our results with three scenarios of level-one KK particle mass spectra. First, we consider coannihilation with all level-one KK particles that carry electroweak, but not color charges. Next, we consider coannihilation including colored particles. In both of these cases, we take all of the level one KK particles (other than the LKP) to have the same mass . Finally, we consider coannihilation with all particles with a mass spectrum derived from the loop corrections from [5]. We believe these three scenarios provide a good cross section of the effects of coannihilation. We also provide formulae for the (co)annihilation cross sections so that any other more complicated spectra could easily be calculated based on our results.

The format of this paper is as follows. In Sec. 2 we discuss KK dark matter including past results and motivate the need for the calculations we carry out. In Sec. 3, we briefly describe the method used to calculate the relic abundance of the KK photon, and the effect of including coannihilating particles in this calculation. In Sec. 4, we consider the effect of coannihilation on the relic abundance, and use this to obtain limits on the mass of the corresponding dark matter candidate, the KK photon. We find mass ranges in three scenarios: (1) when electroweak KK excitations coannihilate, (2) when all KK excitations coannihilate, taking all KK particles (except for the LKP) to have the same mass, and (3) when all KK excitations coanihilate assuming the mass spectrum is determined by the one-loop radiative corrections in [5] with zero matching corrections. Appendices A and B reviews the Lagrangian and Feynman rules in UED. The diagrams and cross sections pertinent to these results are summarized in Appendices C and D.

## 2 KK dark matter

The relic density of the KK photon was first calculated by Servant and Tait [6]. Using their results, one finds that its thermal relic abundance of the KK photon matches the WMAP observations for cold dark matter when the KK photon mass is between about 550 to 850 GeV. The range of mass that they found depended on the relative importance of coannihilation with the level-one KK excitations of the right-handed leptons. This was the only coannihilation channel that was considered in [6]. Several groups have also examined the prospects for detecting KK dark matter directly and indirectly [7].

Coannihilation is likely to occur through other level one KK excitations. Examining a typical spectrum of KK excitations from the one-loop radiative corrections discussed in [5], one finds that the KK excitations of the particles transforming under the electroweak (but not color) groups tend to be within about 10% of the mass of the KK photon. This is near enough in mass that coannihilation with these KK excitations is expected to affect the the thermal relic abundance. We emphasize that these radiative corrections, proportional to the log of the ratio of the cutoff scale to the compactification scale , are merely indicative of shifts in the level one KK particle masses, since the finite matching corrections are unknown. Indeed, if the cutoff scale were not too large, even the KK excitations that transform under SU(3) could also play an important role in the calculation of the thermal relic abundance.

What is the cutoff scale of UED models? Generically, the cutoff scale is where the extra dimensional theory gets strong. In the four dimensional effective theory this can be estimated by including KK excitations into loops, and one obtains typically –. But there have also been recent explicit scattering amplitude calculations that suggest the cutoff scale may be much lower than previously assumed [11]. In any case, it is clear that including the effects of coannihilation for the entire spectrum will allow us to calculate the broadest range of KK photon masses that lead to a thermal relic abundance consistent with the cosmological measurements. We will illustrate the relative importance of different channels by showing the results for specific choices of the level one KK spectrum.

The lower bound on the mass of the KK photon arises by calculating the effects of KK particles in loop corrections to precision electroweak data. This was first done by [3] who found a lower bound of about GeV. As we will see, this bound is not saturated no matter what spectrum of level one KK excitations one takes, and so we do not need to consider it further.

In general, the accuracy we desire requires only tree-level calculations.
However a very recent paper did find an important effect on the relic
abundance of the KK photon [12]. The loop suppressed
operators on the orbifold fixed points lead to couplings between
pairs of level-one excitations and SM particles through
*resonances* of level two (and higher) KK excitations.
The resonance behavior can roughly compensate for the loop suppression,
resulting in a large contribution to the total cross section.
Ref. [12] investigated the
effect of the annihilation
through an channel , and found that including this process leads
to a significant decrease in the relic abundance of .
Including these loop-suppressed effects would make our calculations
prohibitively complicated, so in this paper we neglect them.
Instead, our motivation is orthogonal, namely to
understand the implications of including coannihilation of all
level one KK excitations on the
relic abundance. Ultimately, to obtain the most accurate estimate of the
relic abundance of the KK photon would require taking into
account both the resonance effect as well as the coannihilation
effects that we report in this paper.

## 3 The Relic Density of the KK photon

### 3.1 Annihilation and freeze-out

We now briefly review the standard method for calculating the cosmological relic density of a stable particle [13, 14]. The number density of a particle in the expanding universe is described by the Boltzmann equation

(1) |

where is the thermal average of the total cross section for annihilation of the particle (assumed here to be equal to the total cross section for producing ) times the relative velocity of two particles, and is the Hubble expansion rate of the universe.

We solve the Boltzmann equation to obtain the number density of a massive particle at late times. At early times (, ), is large, and any deviation of from rapidly goes to . At late times (, ), is tiny since the particles are now too massive to be thermally produced. As decreases due to the expansion of the universe, the particles eventually become too dilute to annihilate and freeze out at a constant density per comoving volume. This freeze-out occurs roughly when the Hubble expansion rate overtakes the rate at which the particles annihilate, .

As we will verify later, KK dark matter freezes out at , so that at freeze out , where , and is the number of degrees of freedom of the annihilating particle. In this case, (1) cannot be solved analytically. Instead, the Boltzmann equation must be solved either numerically, or by means of a standard approximation which gives solutions consistent with numerical results to within . Here we shall use the latter approach.

To solve the Boltzmann equation approximately, we begin by making a simplifying change of coordinates. If we assume the universe expands adiabatically, then constant, where is the comoving entropy density given by and is the effective number of relativistic degrees of freedom. In terms of the variables , , and , (1) can be rewritten as

(2) |

where we assume that freeze-out occurs in a radiation dominated epoch, so that . Writing , this becomes

(3) |

where , and .

At early times, the particles track their equilibrium values, so that and . In this case

(4) |

To determine the freeze-out temperature, we use this expression for to solve with , where is a numerical factor whose optimal value is determined by comparison with numerical solutions of the Boltzmann equation (). The freeze-out temperature can be deduced by solving the following equation numerically:

(5) |

At late times, and are both small, and the equation reduces to

(6) |

This equation is separable, and can thus be solved (with boundary conditions again chosen to give the best fit to numerical simulations):

(7) | |||||

Note that because of the factor of appearing in the numerator, this solution assumes that the universe expands adiabatically between freeze-out and today. This is, of course, not quite true, as some standard model particles will subsequently fall out of equilibrium, slightly increasing the entropy per comoving volume. In practice, however, this contribution from known standard model sources is very small, and hence may safely be neglected.

### 3.2 Coannihilation

Different species of interacting particles that have masses nearly degenerate with the KK photon will fall out of equilibrium at the same time and with roughly similar density. Interactions of the form (where and are the KK particles, and are SM particles), converting the relic into other particles of similar mass, occur rapidly. In fact, such processes are much more efficient than the relevant annihilation processes, so that the abundances of the all particles are effectively correlated during freeze-out. The slightly heavier particle species will eventually all decay to the lightest stable particle (in our case, the KK photon) and thus contribute to the relic density. This situation is called coannihilation.

As shown in [14], to compute the relic density and freeze-out temperature with coannihilation, it suffices to substitute an effective cross section for in Eqs. (3.1),(5) above, where

(8) |

The relic abundance depends on a weighted average of the annihilation and coannihilation cross sections of all relevant particles. Thus if the coannihilating particles interact more strongly compared with the annihilating particles (while also being close enough in mass) they can increase the effective cross section and thereby decrease the relic density. Conversely, if they are more weakly interacting, then the coannihilation effects decrease the effective cross section, yielding a larger final value for .

A simple intuitive picture of the effects of including coannihilation comes from noting that

(9) |

The integral over will of course result in a different weight for and wave annihilation, nevertheless it is easiest to simply ignore this difference and approximate . A larger effective cross section results in more annihilation during freeze-out, and thus a smaller relic density.

Furthermore, it is easy to see that the effective cross section is a weighted average of the relevant annihilation and coannihilation cross sections by considering the case where all but the lightest (first) coannihilating particles have the same mass , , and all particles have the same number of degrees of freedom. Then

(10) |

where run over all included particles, for the LKP (all other KK particles), and is the total (co)annihilation cross section for the process Standard Model particles.

This makes it clear that in the limit of a small mass difference and equal numbers of degrees of freedom, is precisely the average of all relevant cross sections while the relic abundance depends only on whether this average is larger or smaller than . If all are equal, then including coannihilation does not alter the relic density; if the weighted cross sections of the coannihilating particles are on average larger (smaller) than that of the original particle, then the relic density will be smaller (larger) when coannihilation is included. This holds irrespective of the value of . However, since enters exponentially into the weights of the coannihilation cross sections, the effects of coannihilation are rapidly suppressed as the relative mass difference increases. Typically for weakly interacting particle dark matter, to , and for the effects of coannihilation are generally negligible. However, for very large coannihilating cross sections, fractional mass differences up to can affect the relic density [14].

## 4 Results

Now we are ready to consider the effect of coannihilation on the relic abundance of the the KK photon. Each coannihilation scenario considered requires several total annihilation cross sections; the relevant Feynman diagrams, together with tables of the formulae of the relevant cross sections, are presented in Appendices C and D.

Certain approximations were made in obtaining our results. Cross sections were computed ignoring all terms of order where is any standard model mass. Consequently, we ignored all Yukawa couplings which are proportional to the corresponding fermion mass. This approximation is well justified (in that including such terms should alter the relic abundance by less than ) for all particles except possibly the top quark, whose mass GeV is close to one half of the precision electroweak lower bound on . In particular, as shown in [12], the top Yukawa coupling can alter the cross section significantly, as it leads to nearly resonant -channel diagrams. Aside from this effect, our results are not expected to be sensitive to the top mass since, as we will see, the masses of the KK photon necessary to explain the observed relic dark matter abundance are well above the mass of the top quark. It addition, we neglected the mixing between and , which are expected to be rather small already for the first KK level (see Appendix A.3). Hence, we take , throughout.

When examining the relic density as a function of mass, a further
simplification is obtained by ignoring the mass-dependence of .
The value of depends very weakly on the mass. Indeed,
from (5) we expect this dependence to be approximately
logarithmic. Typically, over the mass range of to TeV,
varies by about GeV/degree, or less than 10%.^{*}^{*}*The
dependence of on the relative masses of the coannihilating
particles, however, has a
somewhat larger effect, and have been taken into account. This variation
has a small effect on the relic density. This also shows that
the KK dark matter is cold: for all cases considered here,
we find , so that the particles are well approximated
as non-relativistic.

We use couplings and astrophysical parameters that can be obtained from [15]. In particular, we take the electroweak and strong couplings evaluated at (ignoring the renormalization group running of the couplings). In practice, the freeze-out temperature is between about and GeV, depending on which particles coannihilate. The dark matter relic density is taken to be the value from WMAP, [1].

### 4.1 Coannihilation with simplified level-one KK spectrum

To investigate the effects of KK particles coannihilating with the KK photon, we chose two different mass spectra for the coannihilating particles. First, particles in the first KK level are divided into two groups: those with masses approximately , with small, and those that are too heavy to play a role in coannihilation. While this choice of mass spectrum gives a good understanding of the range of possible outcomes of including coannihilating particles, it is also somewhat arbitrary. Thus, we also investigate a mass spectrum based on that derived in [5] in the next subsection.

The first check on our results was to consider coannihilation of the KK photon with just right-handed leptons, where our calculations and numerical results agree with [6]. This provides a non-trivial check on our procedure for calculating the relic density in the Universal Extra Dimensions model.

Figure 1 shows the effect of including coannihilation of the KK photon with leptons (blue), and all electroweak particles including leptons, scalars, and electroweak gauge bosons (red). Here we assume in each case that particles included in the coannihilation have the same mass . The graph shows that coannihilation with leptons, and to a lesser extent with and bosons, tends to increase the relic density for a given mass of the KK photon.

Figure 2 shows the effect of including coannihilation of the KK photon with all level one KK electroweak particles and level one KK quarks (blue), and all particles at KK level one including the KK gluon (red). Again, we assume in each case that all particles included in the coannihilation have the same mass . Clearly, including coannihilation with strongly interacting KK particles decreases the relic abundance for a fixed KK photon mass. The extent of this decrease is one of the most important results of this paper. In particular, if the level one KK particles are highly degenerate to within a few to perhaps 10%, the KK photon mass consistent with the thermal relic abundance that matches the WMAP data could be several TeV.

These results can be more readily understood by examining the relative magnitudes of the cross sections in question, shown in in Fig. 3. Notice first that the coannihilation cross section of any non-colored particle with is smaller than the annihilation cross section of with itself. Thus coannihilation with particles whose self-annihilation cross sections are close to those of tends to decrease the effective cross section, and hence increase the relic abundance for a given mass. This is the case for leptons and scalars. The level one KK electroweak gauge bosons and and all strongly interacting particles have sufficiently large self-annihilation cross sections that they cause an increase in the effective cross section, and decrease the relic abundance. For and this effect is relatively small, and their inclusion does not completely counter-balance the reduction in resulting from including leptons and scalars. In the case of the strongly interacting particles, the relevant cross sections are approximately an order of magnitude larger than those of the non-strongly interacting particles, and the resulting reduction in the relic density is quite dramatic at small .

### 4.2 Coannihilation with the Cheng, Matchev, Schmaltz spectrum

In this section we consider the spectrum that results if one takes the one-loop radiative corrections to the KK masses from [5]. There are several assumptions built into this spectrum. One is that the matching contributions to the brane-localized kinetic terms are assumed to be zero when evaluated at the cutoff scale. Furthermore, the radiatively generated terms are log enhanced by a log of the ratio of the cutoff scale to the mass of the KK excitation. This ratio is also not known, and may be much smaller than previously estimated [11]. We therefore consider a wide range of to illustrate the effects of coannihilation.

First, let us summarize the results of [5] that are used for the masses of the KK excitations in this Section:

(11) |

All of the non-colored KK excitation masses are within about 10% of up to moderately high values of the cutoff scale (), and hence are almost certainly relevant for coannihilation. Conversely, for , the masses of the strongly interacting particles are more than greater than , and thus are less likely to be important for coannihilation. However, as emphasized by [14], particles with sufficiently large cross sections may be relevant to coannihilation even for mass differences of up to of order .

We further simplify the mass spectrum given by (4.2), while ignoring the SM top quark mass, by dividing the KK particles into five mass classes: the KK photon; the KK leptons and scalars; the KK and gauge bosons; the KK quarks; and the KK gluon. (In cases with more than one particle of slightly differing mass, we take the average mass of all of the particles in that class.)

Figure 4 shows the mass that results in a thermal relic abundance consistent with cosmological data, as a function of , given a five class mass spectrum. The upper plot also shows the mass gap for each of the classes. Because is considerably larger than , the masses of the strongly interacting particles increase rather quickly as the cutoff increases. For , nearly all strongly interacting particles have . Masses of the leptons and scalars, however, vary rather slowly with , and remain within of the mass out to values . Thus as the cutoff is increased, the gluons, quarks, and to a lesser extent the and bosons, rapidly become too heavy to play a role in coannihilation, and the relic density is determined by the effective cross section of and the KK leptons and scalars. Thus the mass spectrum, Eq. (4.2), favors a somewhat lower value of GeV to be consistent with the measurements of the dark matter abundance in the Universe.

## 5 Conclusions

We have calculated the thermal relic abundance of the KK photon in the five dimensional UED model for a generalized mass spectrum of level one KK particles. We find that the lowest KK photon mass which could possibly account for the observed dark matter relic abundance is GeV, resulting from including coannihilation with three generations and neutrinos all very nearly degenerate with (). This is consistent with the result found in Ref. [6] updated to reflect the WMAP measurements. Since the radiative mass corrections should be the same for and , a more realistic estimate, given by including all KK leptons, is GeV. This is significantly lower than the lower mass bound of GeV given by including alone. Including level one KK quarks and the KK gluon with masses within about 10% of the mass of the KK photon significantly increases the total effective annihilation cross section. This implies the KK photon mass leading to a thermal relic abundance consistent with WMAP observations is much larger, up to several TeV, see Figs. 2 and 4 for more precise numerical ranges. On face value, such a small separation between the KK photon and the strongly interacting level one KK particles is not expected from the radiative corrections to the masses of the first KK level computed in [5]. However, if the cutoff scale is not much larger than the KK photon mass itself, and thus matching corrections are comparable in size while opposite in sign to compensate, the level one KK spectrum could be much more degenerate. These results show that the range of the KK photon mass is much wider if indeed the mass spectrum is more degenerate than previously thought. Given measurements of the level one KK particle masses, the calculations presented here could be used to find the total effective cross section to verify if the KK photon does (or does not) make up the dark matter density needed to be consistent with WMAP observations.

Note added: During the completion of this work we became aware of an analogous calculation done independently by another group [16]. We have compared extensively the formulas for the annihilation cross-sections involving KK quarks, KK leptons and KK gauge bosons, in the limit of degenerate KK masses, as listed in the Appendix. In all considered cases we found perfect agreement.

## Acknowledgments

We are grateful to K.C. Kong and K. Matchev for extensively comparing their results with ours. We thank G. Servant and T. Tait for discussions. GDK thanks the Aspen Center for Physics for hospitality where part of this work was completed. This work was supported in part by NSERC and by DOE under contracts DE-FG02-96ER40969 and DE-FG02-90ER40542.

## Appendix A Universal Extra Dimensions

Here we provide a brief overview of the Universal Extra Dimension model in five dimensions (for a review, see [17]). In subsequent Appendices we list the Feynman diagrams for (co)annihilation and the cross section results we obtained. The particle content of the UED model is shown in Table 1. The orbifold projects out one helicity of the fermion zero modes, as well as the zero mode of the component of the gauge field. We begin with a brief survey of the mass eigenstates of the theory; the Feynman rules are presented in Appendix B.

field | Even Fields | Odd Fields |
---|---|---|

### a.1 Fermions

The kinetic terms for fermions have the form:

(12) |

where we have ignored SM mass terms. Upon doing the usual Fourier expansion for, for example, the first generation of (left-handed) quarks, and integrating over , we obtain:

(13) | |||||

(We use lower case letters to denote SM particles, and upper case for their KK excitations.) To obtain the full fermion masses, the usual mass terms arising from the Yukawa couplings are added (then diagonalizing the resulting mass matrices).

### a.2 Gauge Bosons

After compactification, the kinetic terms in the gauge boson Lagrangian can be expressed as

(14) |

where is the 4D gauge field, and (to make the analogy with the Higgs mechanism more apparent) , a 4D scalar. Integrating over the fifth dimension coordinate causes all cross-terms between modes of different KK-number to cancel, leaving

(15) |

Expanding the second term, and collecting modes of different KK number, we obtain:

(16) | |||||

This is precisely the Lagrangian for a tower of 4D gauge fields with masses generated by a spontaneously broken symmetry. The scalar fields are eaten by the gauge field in the usual Higgs mechanism to give them mass.

### a.3 Electroweak KK Gauge Boson Mass Eigenstates

There is a tower of KK gauge bosons for each of the SM gauge symmetries. The electroweak gauge boson KK tower, however, is different from the SM zero modes in the admixture of and . This is because the KK and gauge bosons receive different loop corrections to their masses. The actual KK mass eigenstates can be found by diagonalizing the matrix

(17) |

where , are the radiative corrections to the KK level electroweak gauge bosons (precise expressions can be found in [5]). We can re-express this as:

(18) |

where , and . The first contribution to the masses is diagonal in the - basis, and does not affect mixing. The mass eigenstates result from diagonalizing the second matrix; thus the Weinberg angle at each KK level is determined by the relative sizes of and the electroweak masses . Since is proportional to , if , the KK mass matrix is almost diagonal in the - basis. Since is at least 300 GeV [3], the KK Weinberg angle is much smaller than its SM counterpart. In fact, for GeV (which is roughly the lower bound on the mass of ) and , , and to the accuracy required here we can take , and .

### a.4 Scalars in the Electroweak Sector

We have just shown that in the absence of other scalar interactions is the Goldstone boson eaten by in the effective 4D theory. In the weak sector, however, the interactions between and the KK scalar fields mix up the mass eigenstates. Note that this was also discussed in Ref. [18].

To see this, consider the 2-point vertices of the SU(2) gauge bosons. The 5D gauge field strength contributes a term of the form (16). Now we add to this the interactions with the standard model scalar fields:

(19) | |||||

where , are the five-dimensional fields. We now will determine the physical and Goldstone scalars by examining all of the 2-point vertices in this Lagrangian.

After integrating out the dimension, and adding in the gauge boson kinetic term, the relevant piece of the Lagrangian is:

(20) |

where is the standard model Higgs VEV. The terms involving give 2-point couplings between the KK weak gauge boson and the KK excitation of the corresponding Goldstone boson, which we write as . In the case of , these 2-point couplings involve rather than spatial derivatives, and thus become mass mixing terms between and . In addition, both and acquire KK mass from their interactions with , while gets a mass from its interactions with the Higgs VEV. Thus is no longer purely a Goldstone boson, as it has an electroweak scale mass.

A convenient gauge-fixing functional is:

(21) |

After fixing the gauge via , all 2-point interactions between the scalars and gauge bosons of the effective 4D theory are canceled for any gauge fixing parameter . This means that one is free to choose a gauge () in which the Goldstone boson propagator vanishes. For general values of , there are mass mixing terms between and . This leads to mass matrices for the scalars of the form

(22) |

between and the KK excitation of the corresponding SM Goldstone boson. Here and are the masses of the SM and gauge bosons, respectively.

To obtain the physical and Goldstone particles, we diagonalize the mass
matrix.^{†}^{†}†To avoid potential confusion with signs, note that the signs
of the off-diagonal terms in the mass matrix depend on how we define
and .
For the above, we have used the parameterization ; interchanging and in
this convention will change the sign of the off-diagonal terms in the mass
matrix, which introduces a relative sign between the and
components of the physical scalar. In order to ensure that the
cancellations necessary to maintain the unitarity of the theory
occur, care must be taken to use the correct sign
conventions when computing vertex couplings.
We find the mass eigenstates

Clearly is the Goldstone boson, which we can be eliminated from the theory by taking the limit , leaving the physical scalar particle. The fact that the physical scalar is a mixture of and changes its couplings to other particles by terms proportional to the masses of the electroweak gauge bosons. Note that this mixing in the scalars is needed to ensure unitarity of gauge boson scattering is preserved, which we explicitly verified.

Since the component of the physical scalar is suppressed by relative to the component, in practice this mixing can be ignored in all calculations not involving massive external SM gauge bosons.

This completes our discussion of the particle spectrum of the UED model. The KK zero modes are the particles of the Standard Model. At and higher, the effective theory contains massive vector bosons that have eaten the corresponding Goldstone (or a combination of and for the and ). It also contains both helicities of SU(2) doublet and singlet fermions, with the SM helicity even under the action, and the non-SM helicity odd. Finally, the model contains physical scalars, which are the KK Higgs, , and , that are mixtures of with the KK excitations of the SM Goldstone bosons .

## Appendix B Feynman Rules

In this section the Feynman rules relevant for the calculations used in this paper are written. Only the relevant interactions, namely between SM particles and level one KK modes, are shown. In the diagrams we use double and single lines to denote KK particles and SM particles, respectively.

### b.1 Fermion/Gauge Boson Interactions

The fermion interactions of the KK modes differ from those of the standard model due to the vector-like nature of these higher modes, which introduces helicity operators at certain vertices. We find it convenient to work in unitary gauge, so that the Goldstone bosons do not appear as external particles but instead as the longitudinal polarizations of the massive KK gauge bosons. The relevant fermion interactions, after integrating over the fifth dimension, are

(23) | |||

(24) |

leading to the following vertices:

where ; i.e., this vertex is non-chiral as expected.

#### b.1.1 Fermion/Scalar Interactions

The interactions of fermions with scalars proportional to Yukawa couplings are presented for completeness, even though we do not make use of them since we have ignored terms of order . The Yukawa terms result in

(25) | |||

(26) | |||

(27) | |||

(28) |

where () and () are the KK (SM) SU(2) singlet and doublet fermions, respectively. In addition, the interactions of with fermions can be deduced from the gauge boson/fermion interaction Lagrangian above. The only difference is that is odd under the orbifold , and so it couples to the “wrong” (opposite) handedness of the KK fermions. After integrating over the fifth dimension, the interaction terms become:

(29) |

and similarly for , replacing left with right KK fermions. This leads to the Feynman rules: