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Pseudo Goldstone Bosons Phenomenology in Minimal Walking Technicolor

We construct the non-linear realized Lagrangian for the Goldstone Bosons associated to the breaking pattern of SU(4) to SO(4). This pattern is expected to occur in any Technicolor extension of the standard model featuring two Dirac fermions transforming according to real representations of the underlying gauge group. We concentrate on the Minimal Walking Technicolor quantum number assignments with respect to the standard model symmetries. We demonstrate that for, any choice of the quantum numbers, consistent with gauge and Witten anomalies the spectrum of the pseudo Goldstone Bosons contains electrically doubly charged states which can be discovered at the Large Hadron Collider.


I. INTRODUCTION
The main physics objective of the Large Hadron Collider (LHC) experiment is to uncover the mechanism responsible for the Electro Weak Symmetry Breaking (EWSB). One fascinating possibility is that the EWSB has a dynamical origin [1,2] related to a new strongly interacting field theory similar to Quantum Chromo Dynamics. At times this scenario is also referred to as Technicolor (TC). Gauge theories featuring only fermionic degrees of freedom are free from quadratic divergences and therefore offer a natural solution to the standard model (SM) hierarchy problem. Within this scenario, at the EW scale it is not possible to use perturbation theory because the new underlying gauge coupling is large. The strong sector can, however, be constrained using its global symmetries.
We define with G the new global symmetry which breaks dynamically to the stability group H. The minimal and most investigated possibility is to consider the pattern of breaking SU(2) L × SU(2) R → SU (2)  symmetry as well as space-time symmetries, will appear. The relative ordering in mass of the new states will depend on the specific underlying strongly interacting theory.
In [3][4][5][6][7][8][9][10] the reader can find the phenomenological implications for colliders of this pattern of chiral symmetry breaking. However, a generic vector-like, strongly coupled gauge theory will typically feature larger global symmetry groups. In fact, even when considering the most minimal TC theories featuring just two Dirac flavors three distinct patterns of chiral symmetry breaking emerge [11,12]. If the fermions belong to a complex representation of the underlying TC gauge group then the expected pattern is the one above, i.e. SU(2) × SU(2) → SU (2), if the underlying representation is pseudoreal then the unbroken symmetry is SU(4) expected to break to Sp(4), finally if the representation is real then one expects SU(4) to break to SO (4). We say that it is expected to break to a given maximal subgroup since it is not a mathematical proof that it breaks to this subgroup 1 .
In this work, the following pattern of chiral symmetry breaking is considered. This is the first pattern of chiral symmetry breaking for minimal models of TC with a very rich spectrum of Pseudo Goldstone Bosons (PGB)s. Of the 9 PGBs associated, 3 are eaten by the SM gauge bosons while the others are physical and can be observed at LHC. Our main target is to study the phenomenological implications of these states. Furthermore, at least two distinct extensions of the SM of TC type have been constructed potentially underlying this pattern of chiral symmetry breaking. One model is the Minimal Walking TC (MWT) in which the two Dirac fermions transform according to the adjoint representation of the underlying SU(2) TC gauge group [17][18][19][20][21] and the other is the Orthogonal TC [22,23] in which the underlying gauge group is taken to be an orthogonal group and the two Dirac fermions transform according to the vector representation. Both models are expected to be (near) conformal [23][24][25][26][27], a propriety which helps alleviating the tension with potentially dangerously large Flavor Changing Neutral Currents. In fact in [28] it has been argued that it is best if the TC theory is already in the conformal window (i.e. displays large distance conformality) and that the extension of TC needed to endow the SM fermions with a mass should, de facto, perturb the conformal theory away from conformality and together with the underlying gauge theory trigger chiral symmetry breaking. This was dubbed ideal walking [28], which is the paradigm we have in mind here.
We will concentrate on the effective theory consisting of the PGBs and new leptons .
The latter are required by the internal consistency of the theory. The remaining heavy particle spectrum is taken to decouple here.
After introducing the effective Lagrangian for the PGBs and their interactions with the SM fields as well as the new leptons we press on the phenomenological analysis aimed at their discovery at the LHC. We investigate a number of interesting production and decay channels relevant for the LHC phenomenology while discovering, in the end, a golden channel for the discovery of the MWT PGBs.
gauge group there is a definitive Lattice proof [13] that SU(4) breaks to Sp(4). The effective, linear and non-linear, Lagrangians for SU(4) breaking to Sp(4) describing also the interactions with the SM have been constructed in [14][15][16].
In Section II we briefly summarize the MWT model, and provide the low energy effective theory for the PGBs in III . The coupling to the SM matter is introduced in Section IV. The PGB production at the LHC is studied in Section V and the decays in VI.
We investigate the PGB discovery potential and conclude in Section VII.
The PGB phenomenological analysis is instrumental in discovering the specific underlying extension of the SM of TC type. A recent analysis of PGBs stemming from traditional TC models in which the techniquarks carry color has been performed in [29].
We provide a complementary analysis of the phenomenology of PGBs in which the techniquarks do not carry ordinary color and transform according to the real representation of the underlying TC gauge group.

II. MINIMAL WALKING TECHNICOLOR SUMMARY
In the MWT, the extended gauge group is SU(2) TC × SU(3) C × SU(2) L × U(1) Y and the field content of the TC sector is constituted by the techni-fermions, Q, U c and D c , and one techni-gluon all transforming according to the adjoint representation of SU(2) TC .
The model suffers from the Witten topological anomaly [30] which is cured by adding a new fermionic weak doublet L singlet under TC [19]. Furthermore the gauge anomalies cancel when introducing the SU(2) L singlets E c and N c with the hypercharge assignment below 2 : Techniquarks New Leptons The above condensate is invariant under an SO(4) symmetry.

III. MINIMAL WALKING AND ITS NONLINEAR REALIZATION
The symmetry breaking pattern of the MWT model is SU(4) → SO(4). This leaves us with nine broken generators with associated GBs. The resulting low energy effective theory can be organized in a derivative expansion with cut-off scale 4πF, where F is the GBs decay constant. We first introduce the matrix: where Π a are the 9 Goldstone bosons and X a are the 9 broken generators (see Appendix A). U transforms under SU(4) in the following way: The leading term appearing in the Lagrangian is: 3 A detailed discussion regarding the vacuum alignment for MWT can be found in [31].

5
With D the electroweak covariant derivative: where Here g 1 and g 2 are the hypercharge and the weak couplings, respectively. The value of F is fixed in order to reproduce correctly the electroweak symmetry breaking F = 2m W g 2 . The gauging of the electroweak interactions breaks explicitly the SU(4) symmetry group down to SU(2) L × U(1) Y × U(1) V , while the spontaneous symmetry breaking leaves invariant an SO(4) subgroup. The remaining unbroken group is U(1) Q × U(1) V . The U(1) Q factor is the symmetry group associated to the electromagnetism while the U(1) V leads to the conservation of the technibarion number. A simple illustration of the spontaneous and explicit breaking of the SU(4) symmetry is presented in Fig. 1.

CP -Origins
The gauging of the electroweak interacations breaks explicitly the SU(4) symmetry These states acquire a mass because of the gauging of the SM subgroup in SU(4). An explicit calculation of the mass splitting leads to [28] Here g 1 and g 2 are the hypercharge and the weak couplings respectively. With walking dynamics the value of m walk can be even of few hundreds GeV [28].
However, in order to be sure to obtain a phenomenological viable spectrum, it is possible to introduce an extra source of symmetry breaking. A common ETC mass for all the uneaten PNGBs can be provided by adding the following term to (8) with Note that this mass term is not linear in U as is the mass term for the pions in QCD. This

CP -Origins
The gauging of the electroweak interacations breaks explicitly the SU(4) symmetry These states acquire a mass because of the gauging of the SM subgroup in SU(4). An explicit calculation of the mass splitting leads to [28] Here g1 and g2 are the hypercharge and the weak couplings respectively. With walking dynamics the value of mwalk can be even of few hundreds GeV [28].
However, in order to be sure to obtain a phenomenological viable spectrum, it is possible to introduce an extra source of symmetry breaking. A common ETC mass for all the uneaten PNGBs can be provided by adding the following term to (8) with Note that this mass term is not linear in U as is the mass term for the pions in QCD. This is so since it is expected to come from a four-techniquark interaction term, furthermore weak interacations breaks explicitly the SU(4) symmetry s splitting leads to [28] 2 The gauging of the electroweak interacations breaks expl These states acquire a mass because of the gauging of the S explicit calculation of the mass splitting leads to [28] Here The bosons can be classified according to the unbroken group U(1) V × U(1) Q in the following way: Electroweak interactions split the technipions masses according to the following pattern [31]: In models with walking dynamics the value of m walk can be even of few hundreds GeV [31].
Furthermore it is also possible to introduce a common mass term for the Pseudo GBs (PGBs) by adding the following term to the Lagrangian (8): with This term was already added in [21] and it is expected to emerge from a more complete theory of SM fermion mass generation. It is expected to emerge from a four-techniquark interaction term, and preserves SU(2) L × SU(2) R × U(1) V of SU(4), which contains the SU(2) V custodial symmetry group.

IV. COUPLING OF THE MINIMAL WALKING GOLDSTONE BOSONS TO MATTER
A complete Extended TC [32,33] (ETC) model would fix all the interactions between the PGBs and the fermions (both the SM and the new leptons). Here, we are not going to study a specific ETC mode but consider the couplings respecting the SM symmetries.
It will be useful to know the transformation properties of the matrix U with respect to the electroweak gauge group. Under SU (4), the matrix U transforms as a two index symmetric tensor, or in other words U transforms like the irreducible representation 10 of SU (4): Knowing the embedding of the electroweak generators in the SU(4) algebra (see the Appendix A for the details) it is possible to decompose the representation 10 according to the SU(2) L × U(1) Y group: The identification of these representations inside the U matrix is given by: By expanding the exponential in (6) to the second order in the number PGBs fields, we identify the following phenomenologically relevant quantities entering the interaction Lagrangian relevant for the first LHC searches: 8 Depending on the specific choice of the hypercharge assignment y, one can construct different Yukawa-type interactions involving the matrix U. Odd integer values are required for y to avoid stable composite states with fractional electric charge 4 . In general electrically charge stable states are excluded by cosmology, see for example discussion in [34].
Within this setup the allowed Yukawa terms are 5 : S 1 e c e c + h.c.
H c i is defined as H c i ≡ −iσ 2 H * i , and the e c and l symbols stand respectively for SU(2) L singlet and doublets states of the 3 lepton families (e, µ, τ). In the equations (22) Section VI we will further exploit and investigate the rôle of L Πψψ to single out the golden signatures allowing to detect the MWT technipions at LHC.
The allowed values for y can be further restricted by requiring that the lightest state, among the PGBs and the new leptons, is electrically charged and stable. According to this, we discard the cases y = {−5, 3} and the surviving values of y are {−3, −1, 1}.

V. PGB PRODUCTION AT THE LHC
The LHC has pushed particle physics to a new era after starting to explore physics at the TeV scale. In the absence of discovery, the LHC is imposing strong constraints on several BSM models [35,36]. In oder to understand the potential of the LHC to discover or to set limits on the PGBs sector of MWT, it is crucial to know their production and decay mechanisms. This is the goal of this section.
We start by recalling that the U(1) V is a symmetry of the techincolor theory in isolation and transforms the low energy effective fields as follows: Once the TC sector is coupled to the SM matter, and new leptons, such a symmetry in general breaks. However it is straightforward to show that the U(1) V symmetry is accidentally conserved in L U + L 2HDM . This leads to the phenomenological relevant consequence that the PGBs can only be produced in pairs. This is so since they have to be produced either via SM gauge bosons or quarks. These couples only via L U + L 2HDM . This is not the case with other TC models, where a single technipion production is possible, see for example [29,[37][38][39][40][41].
Furthermore the MWT technifermions are colorless suppressing their production compared to TC models with colored techniquarks. In fact, when the technifermions carry ordinary color, also the production of color singlet states is enhanced via loops of technifermions in gluon initiated processes [29]. These observations should help understanding the following MWT PGBs production mechanisms: ♦ Drell-Yan production One possibility to produce a PGB pair is through the SM gauge bosons in the s-channel.
We take the masses of the PGBs to be above the weak gauge boson masses and therefore the intermediate gauge bosons cannot be on-shell. The elementary process is shown in Fig. 2 (a).

♦ Production in association with two jets
This production can happen through a large number of different diagrams; among which Vector Boson Fusion (VBF) is the most important one. The associated Feynman diagram for VBF is shown in Fig. 2 (b). One of the advantages presented by this process is the possibility of tagging the two jets coming from the associated quarks, thus reducing the background.

♦ Gluon fusion
An initial state with two gluons can produce a PGB pair through a top quark loop. The associated Feynman diagram is shown in Fig. 2 (c). The effective Lagrangian describing the gluon fusion process is given in Appendix C. Here we simply derive the coupling between the top-quarks and PGBs. This coupling can be extracted from L 2HDM (22) by expanding H 1 and H 2 to the second order in the PGB fields (see (21)), namely The mass of the top quark is now m t = F(y 1 + y 2 ). Thus, in general, only one of the PGB pairs couples to the top with a coupling proportional to its mass. The relative size of the couplings y 1 and y 2 is unknown and, for simplicity, we choose them to be equal.

FIG. 2:
Feynman diagrams for different production mechanisms.
In order to estimate the production cross sections, we implemented the model into Madgraph 5 [42] using FeynRules [43]. The analyses performed below are done using the 11 CTEQ6L parton distribution functions [44] and we have not implemented any kinematical cuts for the final state PGBs.
In Fig. 3, the inclusive production cross sections for the PGB pairs with opposite charges are shown as a function of the PGB mass. The production cross sections for y = 1 can be obtained from the case y = −1 by interchanging the names of Π UU and Π DD . For heavier PGBs masses the associate production with a jet pair is the dominant mode whereas in the low mass region the proximity of the gauge boson propagator pole enhances the Drell-Yan production.
The cross sections for the production through the W boson exchange are shown in Fig. 4. Notice that we assume in this plot degenerate masses for the PGBs. With non degenerate masses the lines in Fig. 4 would be slightly modified according to specific final state particles.
The production cross section for a pair of PGBs of mass of around 300 GeV is σ ∼ 1 fb. This is much smaller than the single techinpion production of the models investigated in [37][38][39][40][41] where the typical cross section is of the oder of σ ∼ (10 4 − 10 5 ) fb.

VI. PGB DECAY
We begin by recalling that regardless of the value taken for y, i.e.  The mass splitting dictated by the electroweak corrections given in (12) shows that the lightest PGB is the doubly charged state Π DD . For later convenience, we will use the where i, j = e, µ, τ.
In this setup the remaining states Π UD , Π UU and the new leptons, once produced, are forced to decay into the SM particles through the operator mentioned above. (26) and from the requirement of absence of new stable charged particles.
We expect the direct pair production of the Π DD states, and consequent decaying topology to be the most appealing process to study experimentally. When the mass splitting between the PGBs is large (i.e. M W ), also the final states with 5 and 6 leptons arising from the W bremsstrahlung, are competitive and phenomenologically interesting. For these final states, the SM background is negligible (see [46] and references therein).

Case y = ±1
For y = 1 an interesting scenario arises due to the mixing of the new lepton fields and the SM ones which has been partially investigated in [47]. Here we concentrate on complementary signatures and therefore neglect this mixing.
To describe the Π UU , Π DD , and Π UD decays for the y = ±1 cases, it is convenient to re-name these states in terms of their electric charge Q = 2, 1, 0 (e.g. Π 2 , Π 1 , Π 0 ), , and for y = +1 Now, the PGBs interactions with the SM fields, given in (23), read Once more, the cleanest signal comes from Π 2 decaying, at the tree level, into same-sign lepton pairs. Differently from the y = −3 case in which only the lightest technipion could decay into SM fields, here all the states decay directly into SM fields. Here, due to the SU(2) L gauge invariance of the couplings in (32), the lightest state Π 0 can only decay into neutrinos. This gives rise to missing energy in the detector, making the direct detection very difficult. The Π 1 decays through the following tree level decay processes: If Π 1 is pair produced (or is produced in association with Π 2 ) then the variable m T2 (or the transverse mass) can be used to isolate the signal. In other cases the large amount of missing energy will make the signal harder to be observed. More elaborate analysis of these different final states is beyond the scope of this study. We will concentrate, in the following, in analyzing the most appealing scenario corresponding to neat experimental signatures.

VII. DISCOVERY POTENTIAL AND CONCLUSIONS
We have constructed the low energy effective theory for the breaking pattern of SU (4) to SO(4) and specialized it to fit the MWT model. We have then identified the relevant decay modes of the PGBs and are now in a position to conclude this work by confronting theory and experiments.
For every y, the neatest process to be investigated at the LHC involving the PGBs for which we will derive relevant constraints is The flavor structure of the coupling λ i j Π 2 l + i l + j depends on the ETC sector, but given that this sector is unknown we will not assume any specific value for these couplings.
Thus it is possible to have different lepton pairs in the final state. The effect of combining several leptonic final states to the exclusion limits is then compared against the results presented in [48]. According to this study, the limits drawn when the doubly charged PGB is decaying into ee, µµ, eµ, eτ or µτ are comparable. The sensitivity decreases notably if the PGB decays predominantely into four τs. The authors of [45] showed that, for example, in the case of di-muons pairs in the final state, the SM background is negligible and with just few events it is possible to discover the PGBs.
The ATLAS collaboration has studied the production of a doubly charged Higgs boson at the LHC in [49] with 1.6 fb −1 of integrated luminosity. This study was performed by examining the invariant mass distribution of the same sign muon pairs. The doubly charged particle is assumed to decay into two muons with 100% branching fraction. The SM background incorporates non-prompt muons from the pion and kaon decays, and prompt muons from the leptonic decays of WW, WZ and ZZ. Also the production of ttWW contributes to the background. We have used the number of observed events and the number of expected background events reported in table 2 of [49] to calculated the 95% exclusion limit for the PGB production. This yields a lower limit of 286 GeV for the PGB mass with the following acceptance cuts for the muons: The exclusion plot is presented in Fig. 5 for 1.6 fb −1 . We also present the expected 95% confidence level exclusion limit for 5 and 15 fb −1 . From the discussion at the beginning of this section is clear that even if one allows for a more general coupling to electron and muons (but not only to taus) the analysis above is not strongly affected.
We list below, for completeness, several combinations of the LHC total energy and luminosity together with the doubly-charged PGB mass that can give 10 signal events.
Even though the production rates for the PGBs are smaller than the TC models featuring colored techniquarks [37][38][39][40][41] we have shown that LHC can probe a similar mass range.
This is so since in MWT the PGBs signal has negligible SM background compensating for the smaller production rates. For a direct comparison with the early models see [29].
We note that a doubly charged scalar appears as part of the spectrum in several extensions of the SM like the Left-Right symmetric models [50], Little Higgs [51], 3-3-1

Appendix A: SU(4) generators and PGBs quantum numbers
It is convenient to use the following representation of the SU(4) hermitian generators where A is hermitian, C is hermitian and traceless, B = −B T and D = D T . The S are also a representation of the SO(4) generators, and thus leave the vacuum invariant SE+ES T = 0 .
Explicitly, the generators read where a = 1, 2, 3 are the Pauli matrices and τ 4 = 1. These are the generators of SU (2) with Notice that S 4 , S 5 and S 6 are the generators of another SU(2) V algebra and that SO(4) The remaining generators which do not leave the vacuum invariant are and with D 4 = 1 , The generators are normalized as follows Tr S a S b = 1 2 δ ab , , Tr X i X j = 1 2 δ i j , Tr X i S a = 0 .
The electroweak subgroup can be embedded in SU (4), as explained in detail in [15].
The S a generators, with a = 1, .., 4, together with the X a generators, with a = 1, 2, 3, generate an SU(2) L × SU(2) R × U(1) V algebra. This is easily seen by changing genarator basis from (S a , X a ) to (L a , R a ), where with a = 1, 2, 3. The electroweak gauge group is then obtained by gauging SU(2) L , and The gauging of the electroweak group breaks explicitly SU(4) down to H = SU(2) L × U(1) V × U(1) Y . As SU(4) spontaneously breaks to H = SO(4), SU(2) L × SU(2) R breaks to SU(2) V , which acts as a custodial isospin, which insures that the ρ parameter is equal to one at tree-level.
As a consequence of the explicit and spontaneous breaking of SU(4), the unbroken subgroub is given by H ∩ H = U(1) V × U(1) Q . The electromagnetic group is generated by Q = √ 2 S 3 + y S 4 .
We also normalize the U(1) V generators in the following way: In order to couple the PGB to the SM fermions we study the transformation properties of the matrix U. Under SU(4), the matrix U transforms as a two index symmetric tensor, or in other words U transforms like the irreducible representation 10 of SU(4).
It is convenient to introduce the following notation: as well as : with the following form for the projectors P: