(2) ABSTRACT The U (1)D gauge sector containing one dark Higgs boson hD and one dark photon γD may be explored through the decays of the 126 GeV particle discovered at the Large Hadron Collider (LHC), assumed here as the heavier mass eigenstate h1 in the mixing of the standard model h with hD . The various decays of h1 to γD γD , h2 h2 , h2 γD γD and h2 h2 h2 would yield multilepton final states through the mixing of γD with the photon and the decay h2 → γD γD , where h2 is the lighter dark Higgs. Future searches for signals of multilepton jets at the LHC may reveal the existence of this possible dark sector governed simply by the original Abelian Higgs model. Non-Standard Model(SM) massive fermions are also introduced to induce the kinetic mixing between the dark U (1)D and the standard model hypercharge U (1)Y at one loop order. We also compute various three point functions induced by the non-SM fermions at one loop for the neutral gauge and Higgs bosons in the external legs. Experimental constraints on the induced mixing parameter and three point vertices are studied in detail and their implications for future experimental search for a dark U (1)D sector are discussed. Keyword: Quantum Field Theory, particle physics, phenomenology. 2.

(3) CONTENTS. Abstract. 2. Acknowledgment. 5. I. Introduction. 6. II. The Standard Model of Elementary Particles Physics A. The historic motivation of spontaneous symmetry breaking. 8 8. B. Gauge Bosons. 10. C. Scalar Bosons. 10. D. Yukawa Coupling and Fermion Masses. 16. III. Higgs Production and Decays at the LHC. 20. A. Higgs Production and Decays in Standard Model. 21. B. Higgs Production and Decays with LHC result. 30. IV. Brief Introduction of Anomaly. 33. A. Ward–Takahashi Identities. 34. B. Anomaly Cancellation. 41. V. Multilepton Higgs Decays through the Dark Portal. 44. A. SU (2)L × U (1)Y × U (1)D Model. 44. B. Mixing Effects. 46. 1. Higgs Mass Eigenstates and Their Self Interactions. 46. 2. Kinetic and Mass Mixing of the neutral gauge bosons. 47. C. Non-standard Decays of h1. 49. D. Branching Ratios. 52. E. MultiLepton-Jets At LHC. 55. F. Conclusions for Multilepton Higgs Decays. 60. 3.

(4) VI. Implications of a Dark U (1)D Sector with non-SM Heavy Fermions. 62. A. Kinetic Mixing. 62. B. Experiment Data. 63. 1. Fixed Target and Beam Dump Experiments. 63. 2. ATLAS and CMS Results. 64. 3. Higgs and Z boson Decay to Invisible. 65. C. The Model. 66. 1. Scalar and Vector Bosons. 67. 2. Kinetic Mixing and Fermions. 68. 3. Mass Eigenstate. 72. Scalar Mixing. 72. Vector Boson Mixing. 73. D. Phenomenology. 75. 1. Standard Higgs h1 decay modes. 75. h1 → γγ. 75. h1 → ZZ ∗. 77. h1 → Invisible. 79. 2. Standard Z boson decay modes. 80. Z → γh2. 80. Z → γγD. 83. Z → Invisible. 84. E. Discussion. 85. F. Conclusions for Dark U (1)D with non-SM Fermions. 90. Appendix. 92. G. Shift of integration variable for linearly divergent integrals. 92. H. One-Loop 3 point function. 93. References. 100 4.

(5) ACKNOWLEDGMENT I am grateful to my advisor, Professor Tzu-Chiang Yuan, for all the teaching, support and a lot of opportunities provided during my years of M.S.. I truly appreciate your amazing help very much. I would like to express my gratitude to my co-advisor, Professor Chia-Hung Chang who is a good teacher to excite student’s interest. I will miss your course. Lastly, I also would like to express my gratitude to my another co-advisor, Professor Wah-Keung Sze, your teaching is an important property of the student.. Thanks to three Great Teachers.. Thanks to my mother for all the support from I was born. Chia-Feng Chang. 5.

(6) I.. INTRODUCTION. In recent decades, the Standard Model(SM) has been remarkably successful in explaining a huge amount of data collected so far at various high energy particle collider experiments throughout the world. The Large Electron-Positron(LEP) already explored the nature of the gauge interactions with a very high precision [15], leaving little room for theories that differ significantly from the SM at energies below the TeV scale, and the advent of the Large Hadron Collider(LHC) a whole new energy range is opening up to experimental particle physics. The standard model is not perfect. One problem is that the quantum correction to the Higgs mass is not stable; it is quadratically divergent and requires a very fine tuning of parameters. This is known as a fine tuning or naturalness problem in the literature. Another problem is that exist the unexplained hierarchy between the electroweak symmetry breaking scale (∼ 200GeV ) and the fundamental scale of Gravity in four dimensions, the Planck scale (∼ 1019 GeV ). As all the problem, we believe that the SM is a description on a low energy region. Therefore we consider a low energy scale < 10T eV or a scale independent mechanism(e.g. Holdom’s kinetic mixing) in following article. The original Higgs model [1] of spontaneous symmetry breaking involves just one complex scalar field χ and one vector gauge field C. As χ acquires a nonzero vacuum expectation value (VEV), the physical spectrum of this model consists of a massive vector boson γD and a massive real scalar boson hD , and the only interactions between them are of the form hD γD γD and h2D γD γD . The analog of hD in the electroweak SU (2) × U (1) extension [22] of this original model, commonly called the Higgs boson h, is presumably the 126 GeV particle observed at the Large Hadron Collider (LHC) [35, 36]. Is this the whole story? Perhaps not, because the original Higgs model may still be realized physically, but in a sector which connects with the standard model (SM) of particle interactions only through hD − h mass mixing and γD − γ kinetic mixing [5]. If so, the 126 GeV particle may be identified with the heavier mass eigenstate h1 and decays such as h1 to γD γD , h2 h2 , h2 γD γD and h2 h2 h2 would result in multilepton final statesvia γD → ¯ll or h2 → γD γD and then followed by γD → ¯ll, 6.

(7) where h2 is the lighter dark Higgs and l is the SM lepton. By far the extra U (1)D is not only predicted by some phenomenology theories[5, 21, 23– 30], like dark matter or as a bridge to connect the dark matter, but also some Grand Uified Theories(GUT), like E6 or higher rank groups[6]. Furthermore, the new photon is not only of interest since it provides a solution to the discrepancy in the muon(or electron) anomalous magnetic moment between the SM prediction and the experimentally measured value[27, 34], but also is possible that a dark matter (DM) candidate in the extended gauge sector[23– 25, 31, 63–65] if the SM particles don’t have to be charged under the extra U (1)D . The dominant interaction at low energies between the dark photon γD and the SM sector is through kinetic mixing with the U (1)Y gauge boson. Based on idea from [5], the kinetic mixing is via the fermion loop between U (1)D and U (1)Y wave functions(see Sec.VI A). We consider that the fermions probable having not only hypercharge, but also carries weakisospin, and the U (1)D probable is a anomaly U (1) gauge. In this way, the kinetic mixing probable exist some process beyond the original kinetic mixing U (1) model[5], e.g. Z → γD γD , h1 → ZZ ∗ via the fermion loop. We’ll also consider a scalar field hD which do couple to non-SM particles, and also couple to SM-Higgs by scalar potential. The hD could carry a VEV vD to provide a dark photon mass mγD , and the phenomenon of hD is discussed already in Ref.[70].. 7.

(8) II.. THE STANDARD MODEL OF ELEMENTARY PARTICLES PHYSICS. The Standard Model is a renormalizable quantum field that successfully describes all the known particles and the gauge interactions between them (without gravity). Despite its success, there are still many problems to be solved. For example, the theory does not contain any viable dark matter particle that possesses all of the required properties deduced from observational cosmology. The Lagrangian of the Standard Model can be schematically written as LSM = LGauge + LScalar + LY ukawa .. (1). LGauge describes the Yang-Mills gauge interaction of vector bosons and fermions, which incorporates the gauge symmetry of SU (3)c × SU (2)L × U (1)Y . The LScalar is the Lagrangian of the Scalar field that give rises to the Spontaneous Symmetry Breaking Mechanism. The last term LY ukawa is Yukawa coupling between scalar bosons and fermions. Nowadays, many people add neutrino mass terms, which can explain the phenomenon of neutrino oscillation.. A.. The historic motivation of spontaneous symmetry breaking. While the Standard Model has become a major success and almost a reality, historically it’s the various problems gauge field theories encountered when they confront the real world phenomena that motivated great physicists to invent the great ideas that form the Standard Model, especially the idea of spontaneous symmetry breaking. The problems include the following:. • Local SU (2)L × U (1)Y gauge invariance forbids gauge boson masses. When we apply the gauge transformation to the naive mass term of a vector boson, we get: 1 2 1 1 1 1 MB Bµ B µ → MB2 (Bµ + 0 ∂µ γ(x))(B µ + 0 ∂ µ γ(x)) 6= MB2 Bµ B µ 2 2 g g 2 It is evident that the mass term is not gauge invariant and requires a mechanism of Gauge 8.

(9) Symmetry Breaking.. • Local SU (2)L gauge invariance forbids fermion masses. The fermion mass term can be written in terms of the right-handed and left-handed components: ¯ = −mf ψ( ¯ 1 − γ5 + 1 + γ5 )ψ = −mf (ψ¯R ψL + ψ¯L ψR ), −mf ψψ 2 2. (2). Since the gauge symmetry of SM is chiral, ie. φL (is doublet under SU (2)L , I3 = 21 ) and φR (is singlet under SU (2)L , I3 = 0) transform differently under gauge transformation SU (2)L × U (1)Y : i. ψL → ψL0 = e−iω (x)Ti −iγ(x) ψL ψR → ψR0 = e−iγ(x) ψR the fermions mass is not gauge invariant. • Unitarity is violated at high energy. Massive vector boson scattering, for example W W scattering to create ZZ final state, will violate unitarity(S-matrix is not unitary, SS † 6= I) at high energy as σ(W W → ZZ) ∝ k 2 . This energy dependency clearly makes the theory(Fermi’s interaction theory) non-renormalizable. The cross sections are limited by unitarity[2] σ<. 4π , k2. (3). but the cross-sections in Fermi’s interaction theory at high energy limit is σ ∼ G2F k 2 ,. (4). where GF is Fermi’s constant, therefore that unitarity is violating. These three problem motivated physicists to come up with the idea of spontaneous symmetry breaking, especially using a scalar Higgs field. And indeed all three problem are solved and this framework becomes what we know as the Standard Model. 9.

(10) B.. Gauge Bosons. The Lagrangian of the Gauge Bosons describes the fundamental interaction. In the Standard Model, the electromagnetic, weak and strong interactions are described by the gauge theory based on the symmetry group U (1), SU (2) and SU (3). The kinetic energy term is written as: 1 1 1 i Wiµν − Gaµν Gµν LGauge = − Bµν B µν − Wµν a 4 4 4. (5). where Bµν , Wµν and Gµν are the field strengths associated with the various gauge fields. The symmetry group SU (3) contains eight generators, implying the strong interaction is described by eight gauge bosons. Similarly gauged SU (2) contains four gauge boson while U (1) contains one. Each fermion is classified in irreducible representations of the Lie algebra. Their interaction with the various gauge bosons are based on their quantum numbers: Color(C), weak Isospin(I3 ) and Hypercharge(Y). The electric charge is given by the Gell-Mann-Nishijima relation Q = I3 + Y2 . In order for the kinetic energy term to be gauge invariant, we replace the derivatives by the covariant derivatives Y 1 ∂µ → Dµ = ∂µ − igs Gaµ Ta − ig Wµi Ti − ig 0 Bµ 2 2. (6). where the Ta and Ti are respectively the SU (3)c and SU (2)L generators. The kinetic term of fermions can rewrite Lf =. X. iψ¯f γ µ Dµ ψf. (7). f. We list all the fermions in the Standard model and their representation assignment in TABLE.I. It shows that only the left-handed fermions transform under SU (2)L symmetry, while the right-handed fermions are singlet. C.. Scalar Bosons. This subsection briefly review the physics of the Higgs Boson in the Standard Model. The sector is responsible for the Spontaneous Symmetry Breaking (SSB). The idea originated from 10.

(11) TABLE I: Type. Leptons. Quarks. Bosons. a. a. Field. SU (3)c rep.. SU (2)L rep.. I3 (Left). Y (L,R). Charge. Mass[GeV]. νe,L. 1. 2. + 12. -1. 0. < 2 × 10−6. e. 1. 2. − 12. (-1,-2). -1. 2.1 × 10−4. νµ,L. 1. 2. + 12. -1. 0. < 1.9 × 10−4. µ. 1. 2. − 12. (-1,-2). -1. 1.06 × 10−1. ντ,L. 1. 2. + 12. -1. 0. < 1.82 × 10−2. τ. 1. 2. − 12. (-1,-2). -1. 1.78. u. 3. 2. + 12. (+ 13 ,+ 34 ). + 23. (1.5 ≤ m ≤3.0)×10−3. d. 3. 2. − 12. (+ 13 ,− 23 ). − 13. (3.0 ≤ m ≤7.0)×10−3. c. 3. 2. + 12. (+ 13 ,+ 34 ). + 23. 1.28. s. 3. 2. − 12. (+ 13 ,− 23 ). − 13. 9.5 × 10−2. t. 3. 2. + 12. (+ 13 ,+ 34 ). + 23. 173.1. b. 3. 2. − 12. (+ 13 ,− 23 ). − 13. 4.16. h. 1. 2. − 12. 1. 0. ∼ 126. Bµ. 1. 1. 0. 0. 0. Isn’t Mass. Wµi. 1. 3. 1, −1, 0. 0. 1, −1, 0. Eigenstate. Gaµ. 8. 1. 0. 0. 0. 0. A popular denition of the quark mass is the M S mass. For each avour of quarks, this quantity is a well dened, short distance running coupling, provided the relevant scale µ is chosen inside the perturbative region of QCD.. Y.Nambu in 1960. The Lagrangian of Scalar field can be written as: LScalar = |Dµ Φ|2 − V (Φ). (8). V (Φ) = µ2 Φ† Φ + λ(Φ† Φ)2. (9). which potential is. 11.

(12) where µ and λ is parameter of potential V which respect the symmetry of gauge. From TABLE.I, Higgs is a Doublet under SU (2)L symmetry, and it has hypercharge Y = 1, therefore we can write: . φ+ (x). Φ=. (10). 0. φ (x) Let us focus on the potential V, we can easily find the minimum of the potential ∂V = µ2 Φ + 2λΦ(Φ† Φ) = 0 ∂Φ†. (11). If µ2 > 0, Eq.11 has a solution Φ = Φ† = 0, see Fig.1 left. In case µ2 < 0, the potential, as shown in Fig.1, contains the minimum |Φ|2 = −. µ2 v2 = , 2λ 2. λ > 0,. (12). where we take λ > 0 to insure a stable vacuum. This equation implies the vacuum expectation value (vev) of the scalar field is v h0|Φ|0i = √ 2. (13). Fig.1 also implies that at high energy the effect of the nonzero VEV could be ignored VHΦL. VHΦL. -<Φ> ¯. <Φ> ¯. Φ. Φ. FIG. 1: Higgs potential in the case of a real scalar field, depending on the sign of the mass term, µ2 > 0(left) and µ2 < 0(right). and the potential could be approximated as in Fig.1 left, which is symmetrical. At low energy, the effect of the vacuum could not be ignored. As we stated in the last section, local gauge invariance requires the gauge bosons to be massless. But this is not consistent with 12.

(13) FIG. 2: The potential(z direction) for the Higgs field, plotted as function of h(x) and ρ(x). It has a Mexican-hat or champagne-bottle profile at the ground. The figure is from [3].. the experimental observations; W boson and Z boson have been tested by experiment to have the mass. Therefore, they rely on some kind of mechanisms to get the mass. This is called Brout-Englert-Higgs Mechanisms as applied in Standard Model to break Electroweak Symmetry. √ Eq.13 implies the field has a nonzero VEV v/ 2. We can do perturbation expansion around the nonzero VEV and write the field as:. 1 φ0 (x) = √ (v + h(x) + iρ(x)), 2. which h0|h|0i = h0|ρ|0i = 0,. (14). Here h(x) and ρ(x) are the physical fields and we plug the perturbative expansion Eq.14 into the potential. λ V (v, h(x), ρ, φ± ) = λv 2 h2 (x) + λvh(x)(h2 (x) + ρ2 (x)) − (h2 (x) + ρ2 (x))2 + O(φp m) 4. Specifically, the field ρ is massless. It implies a continuous spontaneous symmetry breaking can lead to the existence of massless particles. 13.

(14) Next, we substitute Eq.10 and Eq.14 in kinetic term of Eq.8: 1 Y 1 Y |Dµ Φ|2 = (∂ µ Φ† + ig W iµ Φ† Ti + ig 0 B µ Φ† )(∂µ Φ − ig Wµi Ti Φ − ig 0 Bµ Φ) 2 2 2 2 1 = [(−Bµ g 0 v + gvWµ3 )(−B µ g 0 v + gvW 3µ ) + 2g 2 v 2 Wµ+ W −µ 8 + 2(2∂µ φ+ + ρgWµ+ − I(gvWµ+ + gWµ+ h + Bµ g 0 φ+ + gWµ3 φ+ )) × (2∂ µ φ− + ρgW −µ + I(gvW −µ + gW −µ h + B µ g 0 φ− + gW 3µ φ− )) + (ρ(Bµ g 0 − gWµ3 ) − I(−2∂µ ρ + 2I∂µ h + Bµ g 0 v − gvWµ3 + Bµ g 0 h − gWµ3 h + 2gWµ− φ+ )) × (ρ(B µ g 0 − gW 3µ ) + I(−2∂ µ ρ − 2I∂ µ h + B µ g 0 v − gvW 3µ + B µ g 0 h − gW 3µ h + 2gW +µ φ− ))] In the function we find the terms such as 21 g 0 v∂ µ ρBµ , 12 gv∂ µ ρWµ3 and 2i gv(∂ µ φ− Wµ+ + ∂µ φ+ W −µ ). It means Goldstone boson will mix with gauge boson. The mixing can be simplified by the following transformation: 1 1 Φ → Φ0 = (I − ig ω i (x)Ti − iIg 0 γ(x))Φ, 2 2 1 Bµ → Bµ0 = Bµ + 0 ∂µ γ(x), g 1 0 Wµi → Wµi = Wµi + ∂µ ω i (x) − gjki ω j (x)Wµk , g. (15). here ω(x) and γ(x) is small local function and I is unit matrix. After simplification the Lagrangian can be written as: 1 |Dµ Φ|2 = (∂µ h)2 − 2 1 = (∂µ h)2 − 2 + (Bµ. g2 (v + h)2 Wµ− W +µ + 4 g2 (v + h)2 Wµ− W +µ + 4 . Wµ3 ) . g 02 v 2 8. gg 0 v 2 8. gg 0 v 2 8 g2 v2 8. 1 (v + h)2 (−g 0 vBµ + gvWµ3 )2 8 1 2 h (−g 0 vBµ + gvWµ3 )2 8 . (16). Bµ. . W 3µ. The transformation in Eq.15 is equivalent to the gauge transformation: . φ+ (x). Φ=. . . = φ0 (x). . θ2 (x) + iθ1 (x) √1 (v 2. . → Φ = eθi (x)τ i (x)/v + h(x)) − iθ3 (x) 14. 0 √1 (v 2. (17). + h(x)).

(15) In Eq.15 and Eq.17, the gauge chosen is called unitary gauge. The last term of Eq.16 exist a mixing between B and W 3 and then we can easily diagonal it and find the eigenstate: g02 v2 gg0 v2 2 02 2 Bµ 8 8 = 0 × Aµ Aµ + (g + g )v Zµ Z µ (Bµ Wµ3 ) (18) 8 gg 0 v 2 g 2 v 2 3µ W 8 8 with gWµ3 + g 0 Bµ Aµ = p , g 2 + g 02. gWµ3 − g 0 Bµ Zµ = p g 2 + g 02. It is easy to see that the field Aµ (photon) is massless, while the mass of Zµ is. p g 2 + g 02 v/2. and W ± mass is gv/2. The Goldstone bosons ρ and φ± have been absorbed respectively by the Z and W ± bosons to become their longitudinal components, which means that the number of degree of freedom is still conserved when looking at the Lagrangian before and after the electroweak symmetry breaking. The mixing can be parametrized by a rotation angle θw called Weinberg’s angle g0 , sinθw = p g 2 + g 02. g cosθw = p , g 2 + g 02. (19). In terms of the Weinberg’s angle, the electric chafe can be written as: e = g 0 cosθw = gsinθw = p. gg 0 g 2 + g 02. ,. (20). The parameters of the Standard Model can be measured from(at the tree level) • the fine structure constant α=. e2 g 2 sin2 θw 1 = = 4π 4π 137.03599976(50). Incidentally, the coupling constants is depend on the energy scale, the value of α is 1/(128.95) in MZ scale. • the Fermi weak coupling constant g2 1 GF = √ 2 = √ = 1.16639 × 10−5 GeV −2 2 4 2mW 2v • the Z boson mass MZ =. gv = 91.1882(22)GeV 2cosθw 15.

(16) Through these parameters, we can compute the value of vacuum expectation value(vev), Weinberg’s angle and mass of W boson √ v = (GF 2)1/2 ' 246GeV,. sinθw = 0.2315,. MW = 80.420GeV. and important parameter ρ 2 MW ρ≡ 2 2 =1 MZ cos θw. this value is hardly changed when taking into account radiative corrections. This imposes constraints on any beyond the Standard Model theory. Returning to Eq.9, we can simplify the potential using Eq.17: λ µ2 (v + h(x))2 + (v + h(x))4 2 4 λv 4 λ 4 2 2 3 (21) = λv h (x) + λvh (x) + h (x) − 4 4 √ which show the mass of higgs boson is 2λv. The mass of Higgs boson has to be determined V (v, h(x)) =. experimentally. If the particle recently observed at the Large Hadron Collider turns out to be the Standard Model Higgs boson, that’s mass Mh is around 126 GeV. D.. Yukawa Coupling and Fermion Masses. The fermions acquire masses by Yukawa coupling between the Higgs boson and fermions. The interaction terms between the Higgs field and the fermions can be written as LYukawa = −ydij q¯Li ΦdjR − yuij q¯Li Φc ujR − yi ¯lLi ΦeiR + h.c.. (22). with Φc = iσ2 Φ∗ = . φ0∗. ,. q¯Li = . −. −φ. ui. . . , d. i L. ¯li = L. νei e. ,. (23). i L. These so-called Yukawa couplings have to be added by hand. The matrices −yuij , ydij and yi are arbitrary complex matrices that connect the flavour eigenstates. The thing to note is that the index i and j are the generation number, and thus the Φ could couplez between 16.

(17) different generations. Let’s look how we could treat these quark mixing terms and how we construct that mass of fermion field. Here we spell out the first term of Lagrangian LYukawa explicitly: ydij q¯Li ΦdjR = ydij (ui di )L . φ+. djR. (24). 0. φ + + + φ φ φ 11 yd12 (u d)L yd13 (u d)L d 0 0 yd (u d)L φ0 R φ φ 21 + + + φ φ φ 22 23 yd (c s)L yd (c s)L = yd (c s)L sR 0 0 0 φ φ φ 31 + + + yd (t b)L φ 0 yd32 (t b)L φ 0 yd33 (t b)L φ 0 bR φ φ φ . After spontaneous symmetry breaking(see Eq.14), the following mass terms for the fermion fields arise: v v v v i v LY ukawa = −ydij dL √ djR − yuij uiL √ ujR − ye eL √ eR − yµ µL √ µR − yτ τ L √ τR 2 2 2 2 2 + h.c. + Interaction terms, i. = −Mdij dL djR − Muij uiL ujR + h.c. − Mlii e¯i ei +Interaction terms, with Mdij. ydij v = √ , 2. yuij v Muij = √ , 2. ye v ye v ye v Ml = diag( √ , √ , √ ), 2 2 2. (25). The matrix of quark mass is non-diagonal. We can use the unitary matrix V (VL† VL = 1) to diagonal the matrix of mass: Mddiag = VLd Md VRd† Mudiag = VLu Mu VRu†. (26). The mass of quark can be written as i. ij j i ij j LQuark Mass = −dL Md dR − uL Mu uR + h.c. i. = −dL VLd† VLd Mdij VRd† VRd djR − uiL VLu† VLu Muij VRu† VRu ujR + h.c. i0. 0. 0. 0. i0. 0. 0. 0. = −dL (Mdij )diag djR − uiL (Muij )diag ujR + h.c. = −d (Mdij )diag dj − ui (Muij )diag uj 17.

(18) where in the last line the chiral term is simplify by Eq.2, and the matrices V have been absorbed in the quark states like as 0. 0. 0. 0. VLd djL = djL , VRd djR = djR , VLu ujL = ujL , VRu ujR = ujR ,. (27). 0. where the dj is mass eigenstate. Next the interaction between the fermion and Higgs boson can be written. Back to Eq.24, we can find the interaction between the di , uj and φ± , for example ydij u¯iL φ+ djR + h.c.. (28). As before, we transform the quark state to mass eigenstate by unitary matrix Eq.27, and substitute Eq.25 and Eq.26 into 28, and then we have √ 2 + i0 u ij d† j 0 + i ij j φ u¯L yd dR + h.c. = φ u¯L VL Md VR dR + h.c. √v 2 + i0 u d† diag d d† j 0 φ u¯L VL VL Md VR VR dR + h.c. = √v 0 0 0 2 + i0 † = φ u¯L VCKM Mddiag,i j djR + h.c. v. (29). and where we define . VCKM = VLd VLu† ,. and VCKM. Vud Vus Vub = Vcd Vcs Vcb Vtd Vts Vtb. From the definition of VCKM and Eq.29, fro example, the transition from a up quark to an ∗ strange quark is described by Vus , whereas the transition from an strange quark to an up. quark is described by Vus . Remember φ± is absorb to W ± boson as we have discussed in the last chapter. Hence the interaction also involves W ± boson, therefore we can draw the Fig.3 The CKM name from Cabibbo-Kobayashi-Maskawa which allows for CP-violation in the SM. Incidentally the leptons exist similar matrix, only if the yukawa coupling between Higgs and leptons is non-diagonal. This matrix is known as the Pontecorvo-Maki-NakagawaSakata(PMNS) matrix and that’s numeric is almost diagonal.. 18.

(19) FIG. 3: Flavour violation interaction between strange quark, up quark and W ± boson, which coupling proportional to CKM matrix element Vus .. 19.

(20) III.. HIGGS PRODUCTION AND DECAYS AT THE LHC. The Standard Model of particle physics developed about 40 years ago has been a major success. With the prediction of the neutral currents and the masses of the heavy vector bosons some of the major elements of the theory were demonstrated. Until recently observed a Higgs-like particle at the Large Hadron Collider and many physicist work together to find the last piece of the puzzle in Standard Model, which is exciting. On the level of precision physics, the last many years of data produced by ATLAS and CMS, the two main experiments at LHC, at the 7-8 TeV scale in proton-proton collisions, has shown that the standard model also describe the different radiative corrections with remarkable accuracy but there are at present the experimental results which are exist a few not satisfactory described by the Standard Model, that will be introduced in this section.. (a)ggF. (c)W H and ZH. (b)V BF. (d)tt¯H. FIG. 4: The most important processes for Higgs production at hadron colliders. Gluon fusion (a), vector boson fusion (b), Associative production with W ± and Z (c) and an example of the diagrams having associative production with a top quark pair (d). 20.

(21) A.. Higgs Production and Decays in Standard Model. The dominating Higgs production mechanism at the LHC is the gluon fusion process, for all possible Higgs masses, as shown in Fig4-(a). Other processes (with their Feynman diagrams in Fig.4) are also of interest because they provide special signatures for the identification of the Higgs boson. In this subsection, we review the Higgs production and decay process. • Higgs Production via Gluon Fusion1 The amplitude of the gluon fusion (see Fig.5) can be calculated using the dimensional regularization scheme with the dimension d = 4 − 2: The expression of the amplitude (except. FIG. 5: The Feynman diagrams for gluon fusion, where a,b is SU (3)c color index. for polarization vectors) is Z i dd p i i a ab 2 b igs t γµ yt iMµν = −µ tr igs t γν (2π)d p/ − mt p/ + k/ − mt p/ − k/ − mt. (30). Here yt is Yukawa coupling (see last section). Fig.5 indicates the amplitudes observe a symmetry when we switch k1,µ,a ↔ k2,ν,b . Since the amplitude is invariant under this transform, so we have ba iMab µν = iMνµ;cross .. Next we arrange the amplitude in the following form Z dd p Nµν ab 2 ab 2 , iMµν = −iyt C(r)gs δ µ d (2π) D0 D1 D2 1. Thanks Yoshio Kitadono for teaching. 21. (31). (32).

(22) where tr(ta tb ) ≡ C(r)δ ab , D0 ≡ p2 − m2t ,. (33). D1 ≡ (p + k1 )2 − m2t ,. D2 ≡ (p − k2 )2 − m2t ,. Nµν =tr[(p/ + k/ − mt )γµ (p/ + mt )γν (p/ − k/ + mt )].. (34) (35). Feynman parameters for denominators can be introduced: 1 = D0 D1 D2. Z. 3. d xi δ(1 −. 3 X. xi ). i=1. Γ(3) , − D]3. [l2. l = p + x2 k1 − x3 k2 , D = m2t − 2x2 x3 k1 · k2 = m2t − sx2 x3. (36) (37) (38). 2 = 0 for the initial gluons and s equal to 2k1 · k2 . where we used the on-shell conditions k1,2. We calculate the numerator by Dirac algebra Nµν =tr[(p/ + k/ − mt )γµ (p/ + mt )γν (p/ − k/ + mt )] =4mt [4pµ pν + gµν (m2t − p2 − k1 · k2 ) + k1ν k2µ − k1µ k2ν ],. (39). the last term k1µ k2ν vanish by orthogonality relations between the gluon momenta and their polarization vectors k1 · (k1 ) =0,. (40). k2 · (k2 ) =0.. (41). We then substitute p → l −x2 k1 +x3 k2 into Eq.39, and restrict the terms by gauge invariance and rewrite the above equation as Nµν =4mt [4pµ pν + gµν (m2t − p2 − k1 · k2 ) + k1ν k2µ ] 4 =4mt [(1 − 4x2 x3 )(k1ν k2µ − k1 · k2 gµν ) + gµν (m2t − 2x2 x3 k1 · k2 ) − gµν l2 (1 − )] d 2 2 4 =4mt [(1 − 4x2 x3 )(k1ν k2µ − k1 · k2 gµν ) − gµν (l − D) + gµν l ( )] (42) d 2. where we used lµ lν = gµν ld in the loop integral for l. We perform the loop integrals with the formula from Appendix and found that the last two terms in Eq.42 cancel each other. 22.

(23) Finally the amplitude becomes iMab µν. gs2 = 4mt C(r)yt δ ab (gµν k1 · k2 − k1ν k2µ ) 2 (4π). Z. 1 3. d xδ(1 − 0. 3 X. xi ). i=1. 1 − 4x2 x3 D. We focus on the integral and define the I function as Z 1 3 X f (x1 , x2 , x3 ) 3 I[f (x1 , x2 , x3 )] ≡ d xδ(1 − xi ) , D 0 i=1. (43). (44). consider I[D] 1. Z. 3. d xδ(1 −. I[D] = 0. 3 X. Z. 1. 1−x1. dx1. xi ) = 0. i=1. Z 0. 1 dx2 = , 2. (45). we can decompose I[D] as 1 I[D] = m2t I[1] − sI[x2 x3 ] = , 2. (46). which is already solved, following expression for a amplitude of gluon fusion 4m2h s gs2 8mt ab C(r)y δ (g k · k − k k )(1 + (1 − ) I[1]) t µν 1 2 1ν 2µ (4π)2 s s 2 gs2 8mt C(r)yt δ ab (gµν k1 · k2 − k1ν k2µ )[1 + (1 − τ )f (τ )] = (4π)2 s. iMab µν =. (47) (48). where τ is 4m2 /s and 2 1 −1 sin ( √τ ) f (τ ) = 2 1 τ + ln( ) − iπ − 4 τ−. (1 ≤ τ ), (49) (0 < τ < 1),. with τ≡. 4m2t , s. √ 1 τ± ≡ (1 ± 1 − τ ) 2. (50). We rewrite the amplitude by yt = −imt /v, C(r) = 1/2 and s = m2h , and add the cross term by Eq.31. The final amplitude for gg → h (except for polarization vectors) equals ba ab iMab µν (gg → h) ≡ iMµν + iMνµ;cross. =. −igs2 2τ ab δ (gµν k1 · k2 − k1ν k2µ ) [1 + (1 − τ f (τ ))] (4π)2 v 23. (51) (52).

(24) The squared amplitude with spin and color averaged) XX 1

(25)

(26) 2 1 µ ν ∗ρ ∗σ

(27) M

(28) = Mab · Mab ρσ · (k1) · (k2) · (k1) · (k2), Nc2 − 1 a,b pol 22 µν =. αs2 m4h |2τ [1 + (1 − τ f (τ ))]|2 . 1024π 2 v 2. (53) (54). Note the αs is function of renormalization scale, αs =. 4π 1 , β0 ln Q22. 2 β0 = 11 − nf , 3. Λ. (55). We then plug in the one particle phase space formula: σ(gg → h) =. αs2 m2h |2τ [1 + (1 − τ f (τ ))]|2 δ(s − m2h ), 2 1024π v. (56). The cross section of gluon fusion in proton level can be obtained by taking into account the gluon distribution g(x, µ2F ) in proton. The leading order(LO) cross section for σ(pp → h) can be obtained by the following formula: Z 1 Z 1 σLO (pp → h) = dx1 dx2 g(x1 , µ2F )g(x2 , µ2F )σ(gg → h), 0. (57). 0. where x1,2 is the momentum fraction of gluon in proton, µF is the factorization scale, and now the s in Eg.50 needs to be replaced by the total energy of proton system(sall ) while the s in Eq.56 is the total energy of parton subsystem(sggh ). The two kinetic parameters have a relation sggh = x1 x2 sall Using this relation, we have Z 1 Z 1 Z 1 Z 1 2 dx2 δ(x1 x2 sall − m2h ) dx1 dx2 δ(sg gh − mh ) = dx1 0 0 Z0 1 Z0 1 1 m2h = dx1 dx2 δ(x2 − ) x1 sall x1 sall 0 0 Z 1 1 = dx1 x1 sall 4τ hence, we can rewrite the Eq.57 Z 1 4τ α2 m2h 1 σLO (pp → h) = dxg(x, µ2F )g( , µ2F ) s |2τ [1 + (1 − τ f (τ ))]|2 2 xs 4τ x 1024π v all x 24. (58). (59) (60) (61). (62).

(29) We use the gluon parton distribution function(PDF) g(x, Q2 ) given by Gluck, Reya and Vogt(GRV)[83], and the PDF code from [84]. The result of numerical calculation is shown in TableII.. TABLE II: we chose the parameter: sall = 14TeV, µF = Q = mh , Λ = 0.2GeV, nf = 5, mt = 175GeV mh [GeV]. 124.5 134.5 144.5 154.5 164.5. σLO (pp → h)[pb] 21.6 18.6 16.2 14.4 12.5. The PDF is given as. g(x, Q2 ) =[x0.558 (4879s − 1.383s2 + 25.92x − 28.97sx + 5.596s2 x − 25.69x2 + 23.68sx2 r 1 1 2 − 1.975s2 x2 ) 1.218 + s0.558 exp(−0.595 + 2.138s + 4s1.218 ln )](1 − x)2.537+1.718s+0353s , x x with. s ≡ ln. ln[Q2 /0.2322 ] . ln[0.252 /0.2322 ]. (63). • Higgs Decay in Standard Model Standard Model Higgs can decay into a fermion−antifermion pair. As a general rule, the Higgs is more likely to decay into heavy fermions than light fermions because the mass of a fermion is proportional to the strength of its interaction with the Higgs (see Eq.25),In the following, we calculate the higgs decay. The matrix element of higgs decay to fermion pair equals. −iM = u¯(k1 ) 25. imf v(−k2 ) v. (64).

(30) and the matrix element squared is M2 = = = = =. m2f X v¯s (−k2 )us1 (k1 )¯ us1 (k1 )vs2 (−k2 ), v 2 spin 2. (65). m2f tr [¯ v (−k2 )u(k1 )¯ u(k1 )v(−k2 )] , v2 m2f tr(k/1 + mf )tr(−k/2 − mf ), v2 m2f −4k1 · k2 − 4m2f , 2 v m2f [2m2h − 8m2f ] v2. (66) (67) (68) (69). We can then write down the decay rate of Higgs Boson to fermion pair, following the formula of phase space: Nc mh m2f βf3 , Γ(h → f f¯) = 8πv 2. s with. βf =. 1−. 4m2f m2h. (70). where Nc is color number of fermion. Next, we focus on the process of Higgs Boson decay to gauge bosons. The decay width of higgs to γγ is All Gf m3h α2 X √ Γ(h → γγ) = |Q2i Xi |2 , 2 16π 8π 2 i=F,W. (71). with XF = −2τf [1 + (1 − τf )f (τf )],. (72). XW = 2 + 3τW + 3τW (2 − τW )f (τW ),. (73). where we define 4m2f τf ≡ 2 , mh. τW ≡. 4m2W , m2h. (74). i indicates all possible diagrams, and the Qi is the charge of the particle in the loop. The decay width of Higgs boson to Zγ(The formula from [78]) 3 X GF m3h α2 m2Z IW 2 Qf gf √ Γ(h → Zγ) = 1 − | N I + | c F m2h sinθW cosθW tanθW 4π 2 16π 2 f 26. (75).

(31) FIG. 6: The Feynman diagrams for higgs decay to vector bosons in loop level. with 8m2f m2Z 2 2 2 2 2 2 ) − B (m , m m ) , m , m IF = 2 B (m 0 0 f f h Z f f (mh − m2Z )2 4m2f − 2 −2 + (−4m2f + m2h − m2Z )C0 (0, m2Z , m2h , m2f , m2f , m2f ) , 2 mh − mZ 2 m2Z IW = − mh (1 − tan2 θW ) − 2m2W (−5 + tan2 θW ) ∆B0 2 2 2 (mh − mZ ) 1 [m2 (1 − tan2 θW ) − 2m2W (−5 + tan2 θW ) − 2 mh − m2Z h + 2m2W (−5 + tan2 θW )(m2h − 2m2W ) − 2m2Z (−3 + tan2 θW ) C0Zh ]. (76). (77). and we defined the ∆B0 ≡ B0 (m2h , m2f , m2f ) − B0 (m2Z , m2f , m2f ),. C0Zh ≡ C0 (0, m2Z , m2h , m2f , m2f , m2f ). (78). where the B0 and C0 function is Passarino-Veltman function which we defined in Appendix. The Feynman diagrams of these Higgs decay to vector bosons in loop level are show in Fig.6. The contribution of W ± boson loop(right) is five times bigger then fermions(left) loop contribution. The decay width of Higgs Boson to gluon pair is similar to Eq.51. Finally, we consider the decay modes of Higgs to W ± boson pair and Z boson pair in on-shell case. We use the Feynman Diagrams in Fig.7. The amplitude matrix element of Higgs decay to W ± is iMh→W W = igmW gαβ α (k1 )β (k2 ) 27. (79).

(32) FIG. 7: The Feynman diagram for higgs decay to W ± pair. and the amplitude squared XX spin pol. ! k2β k2ν k1α k1µ βν αµ −g + 2 |Mh→W W | = (gmW ) gαβ gµν −g + 2 mW mW (k1 · k2 )2 = (gmW )2 2 + m4W 2. 2. where we fixing the Feynman t’Hoot gauge in first line, and in on-shell particle, we have 1 (k1 + k2 )2 − k12 − k22 2 1 = (m2h − 2m2W ), 2. k1 · k2 =. We then plug in the phase space of two body decay and write s 2 3 g mh 4m2W 4m2W 3m4W Γ(h → W + W − ) = 1 − (1 − + ). 64π m2W m2h m2h m4h. (80) (81). (82). Now for the decay of Higgs to ZZ, the formula is similar except that we have to divide it by a factor of 2 by two identical particles in the final state. s 2 3 1 g mh 4m2Z 4m2Z 3m4Z Γ(h → ZZ) = 1 − (1 − + 4 ). 2 64π m2W m2h m2h mh. (83). The decay width of formula of Higgs decay to W W ∗ and to ZZ ∗ modes can be write down by2 3g 4 mh F (mW /mh ), 512π 3 sin2 θW + 160 sin4 θW g 4 mh 7 − 40 3 9 Γ(h → Z ∗ Z) = F (mZ /mh ), 2048π 3 cos4 θW. Γ(h → W ∗ W ) =. 2. (84) (85). The formula refer to Higgs hunter and the detail of calculation can be found in [78] and https://twiki.cern.ch/twiki/bin/view/LHCPhysics/CrossSections. 28.

(33) where F (x) ≡ − |1 − x2 |(. 47 2 13 1 x − + 2) 2 2 x. + 3(1 − 6x2 + 4x4 )|ln(x)| +. (86) 3(1 − 8x2 + 20x4 ) −1 3x2 − 1 √ cos ( ) 2x3 4x2 − 1. (87). In this section we have considered all possible process of Higgs decay in Standard Model. We can draw a figure of the branching fractions of the various Higgs Boson decay channels (see Fig.8). In recent years, the experimental data are gradually testing the theoretical calculation. We will discuss the experiment result in next subsection. bb. 57 % ΓΓ, ZΓ ZZ cc. 0.4 % 3% 3%. ΤΤ. 6%. gluon pair. 9%. 21%. W-W+. FIG. 8: Left, the most important decay branching fractions for the decays of Higgs boson. Right, the decay of a Higgs at 125 GeV is proportioned into the following channels.. 29.

(34) B.. Higgs Production and Decays with LHC result. Two beams of particles are accelerated to very high energies and allowed to collide within a particle detector to produce Higgs bosons. Although rarely, a Higgs boson could be created fleetingly as part of the collision products. Because the Higgs boson decays very quickly(life time ∼ 10−22 s), the decay occurs at the collider center, and that’s why particle detectors can’t detect Higgs boson directly. Instead the detectors register all the decay products (the decay signature) and from the data the decay process is reconstructed. All possible decay products move through the detector and are measured in different parts(see Fig.9). If the observed decay products match a possible decay process (known as a decay channel) of a Higgs boson, this indicates that a Higgs boson may have been created. In practice, many processes without involving Higgs Bosons may produce similar decay signatures. In this subsection, all of data and figure are from CERN’s web[4].. FIG. 9: The above generic detector sandwich in the massive detectors shows charged particles of electrons, muons, protons, and pions leaving tracks in the inner silicon tracking chambers. Next, light electrons and photons make showers leading to their total energy measurement in the electromagnetic calorimeter. Contains interesting thing here, for example if some kind of invisible particle will decay to electrons but its life time is so long(path longer than detector) that it can’t be measured. Therefore may exist some ”special” particle here but we can’t find it.. 30.

(35) H (N. NLO. +NN. LL. 10. QC. D+. NLO. EW. ). pp → qqH (NNL OQ CD + pp pp NLO → → EW) W ZH H (N (N N NL LO O QC QC pp D → D + ttH +N (N LO NLO LO EW EW QC ) ) D). 1. -1. 10. 1. LHC HIGGS XS WG 2013. s= 8 TeV. Higgs BR + Total Uncert. pp →. LHC HIGGS XS WG 2012. σ(pp → H+X) [pb]. 102. WW bb. 10-1. gg. ττ. ZZ. cc. 10-2 γγ. Zγ. 120. 140. 10-3 µµ. -2. 10. 80 100. 200. 300. 400. 1000 MH [GeV]. FIG. 10: The production cross section of the standard model Higgs boson.. 10-480. 100. 160. 180. 200. MH [GeV]. FIG. 11: The branching fractions of the standard model Higgs boson.. TABLE III: The Standard Model Higgs boson production cross sections(in pb) √ and Mh = 125GeV in pp collisions, as a function of the center of mass energy, s. √. s(T eV ) 1.96 7 8 14. ggF. V BF. WH. ZH. tt¯H. total. 0.95+17% −17% 15.1+15% −15% +15% 19.3−15% 49.8+20% −15%. 0.065+8% −7% 1.22+3% −2% +3% 1.58−2% 4.18+3% −3%. 0.13+8% −8% 0.58+4% −4% +4% 0.70−5% 1.50+4% −4%. 0.079+8% −8% 0.33+6% −6% +6% 0.41−6% 0.88+6% −5%. 0.004+10% −10% 0.09+12% −18% +12% 0.13−18% 0.61+15% −28%. 1.23 17.4 22.1 57.0. The Fig.10 and Table.III summarizer the data of Higgs boson production cross sections for Higgs mass of 125GeV. The process ggH which accounts for 87.3% the production ratio is dominating, and the Fig.11 and Table.IV describe the Higgs boson decay data of LHC, and then we can compare the result of last subsection then we can find that the result is a similar(almost same) as Standard Model prediction. But still exist a few different part here. The Fig.12 show the results for the measured probability (or cross section) for production of the Higgs in several channels as a ratio to that predicted in the standard model. If the new resonance is only the standard model one, all of the ratios should eventually be 1. In the ATLAS figure, +1 on the x-axis is the dashed line. In the CMS figure, 1 is slightly above the average of 0.88 which is the black line, and the green band is the error of plus or minus 0.21. As we can see, the process of Higgs → 4-lepton jets and Higgs → γγ is more than 31.

(36) TABLE IV: The branching ratios and the relative uncertainty [1201.3084] for a SM Higgs boson with mh = 125GeV Decay channel Branching ratio Rel. uncertainty H → γγ. 2.28 × 10−3. H → ZZ ∗. 2.64 × 10−2. H → Zγ. 1.54 × 10−3. H → W ± W ∓∗. 2.15 × 10−1. H → b¯b. 5.77 × 10−1. H → µ+ µ −. 2.19 × 10−4. H → τ +τ −. 6.32 × 10−2. +5.0% −4.9% +4.3% −4.1% +9.0% −8.9% +4.3% −4.2% +3.2% −3.3% +6.0% −5.9% +5.7% −5.7%. Standard Model, on the other hand the b¯b finl state process is less than Standard Model, those may be would signal new physics as well. The section V is rely on this data to evolved.. 32.

(37) FIG. 12: The measurements looking for signals of a Higgs particle at ATLAS (left) and CMS (right). For each process, the dot is the measurement of the rate relative to the Standard Model expectation, and the horizontal bar gives the one-standard-deviation uncertainty. Note all measurements are within two standard deviations of expectations and the ATLAS average ratio to the standard model called µ is 1.3 ± 0.3. The CMS average is 0.88 ± 0.21 at what is called one standard deviation. IV.. BRIEF INTRODUCTION OF ANOMALY. Symmetries play an essential role in general and in particle physics. A symmetry of the classical action is a transformation of the fields that retains the action invariant. Standard examples are Lorentz, or more generally Poincar´e transformations, and gauge transformations in gauge theories. One must then ask whether these symmetries are still valid in the quantum theory.3 In the functional integral formulation of quantum field theory, symmetries of the classical action are easily seen to translate into the Ward identities for the correlation functions or the Slavnov-Taylor identities for the quantum effective action. An important assumption in the proof is that the functional integral measure also is invariant under the symmetry. If this is not true, these Slavnov-Taylor or Ward identities are violated by a so-called anomaly. 3. Almost the depiction and formulae are learned by [57, 97–99]. 33.

(38) A.. Ward–Takahashi Identities. Divided into four subsections, Ward-Takahashi identity for the vector current, Current divergences from the equation of motion and Anomaly Ward-Takahashi identity for Abelian field.. • Ward-Takahashi identity for the vector current We take a simple model to introduce the Ward-Takahashi identity for the vector current, the φ4 theory with a U (1) gauge symmetry, m2 2 λ 4 1 φ − φ, L = D†,µ φ† Dµ φ − 2 2 4. (88). Dµ = ∂µ + ieAµ ,. (89). with. and we recall the quantum field theory, for Noether’s Theorem, we have scalar current form Eq.88 Jµ = i[(∂µ φ† )φ − (∂µ φ)φ† ],. (90). [∂0 φ† (x, t), φ(x0 , t)] = −iδ 3 (x − x0 ),. (91). and the canonical commutators. then leads to the commutators [J0 (x, t), φ(x0 , t)] = [∂0 φ† (x, t), φ(x0 , t)]φ(x, t) = δ 3 (x − x0 )φ(x, t) [J0 (x, t), φ(x0 , t)† (x, t)] = − δ 3 (x − x0 )φ(x, t)† (x, t). (92) (93) (94). Consider the three-point Green’s function given in Fig.13. Z Gµ (p, q) =. d4 xd4 y e−iq·x−ip·y h0|T (Jµ (x)φ(y)φ† (0))|0i. 34. (95).

(39) FIG. 13: The Green’s function of two scalar fields coupled to a vector current. where T is ”time-ordering” symbol, and then we make the standard current-algebra manipulation µ. Z. d4 xd4 y e−iq·x−ip·y ∂xµ h0|T (Jµ (x)φ(y)φ† (0))|0i. Z. d4 xd4 y e−iq·x−ip·y ∂xµ h0|T ({Jµ (x)φ(y)θ(x0 − y0 ). q Gµ (p, q) = − i =−i. + φ(y)Jµ (x)θ(y0 − x0 )}φ† (0))|0i Z =−i. d4 xd4 y e−iq·x−ip·y {h0|T (∂ µ Jµ (x)φ(y)φ† (0))|0i + h0|T (θ(x0 − y0 )[J0 (x), φ(y)]φ† (0))|0i + h0|T (θ(x0 )[J0 (x), φ† (0)]φ(y))|0i},. the first term on the right-hand side vanishes by current conservation, ∂ µ Jµ = 0, and the other terms can be simplified by using Eq.92 Z µ q Gµ (p, q) = − i d4 x e−i(p+q)·x h0|T (φ(x)φ† (0))|0i Z + i d4 y e−ip·y h0|T (φ† (0)φ(y))|0i, the right-hand side is just the propagators for the scalar field Z ∆(p) ≡ d4 x e−ip·x h0|T (φ(x)φ† (0))|0i,. (96). thus, we rewrite −iq µ Gµ (p, q) = ∆(p + q) − ∆(p), which is an example of vector-current Ward-Takahashi identities.. 35. (97).

(40) • Current divergences from the equation of motion The relationship between symmetries and conservation laws in classical field theory, summarized in Noether’s theorem. Let us consider a vector field Vµ and an axial field Aµ , and then start from a sample Lagrangian ¯ ∂/ + V ¯ / + Aγ / 5 )ψ − mψψ, L(Vµ , Aµ ) = ψ(i. (98). In terms of left and right handed fields we have / L ψL + ψ¯R iD / R ψR − m(ψ¯L ψR + ψ¯R ψL ), L(Vµ , Aµ ) = ψ¯L iD. (99). as we known, several non-Abelian currents are constructed ¯ µ T i ψ, Jµi =ψγ. (vector). (100). ¯ µ γ5 T i ψ, Jµ5,i =ψγ. (axial). (101). ¯ 5 T i ψ, Jpi =ψγ JµL,R,i =ψ¯L,R γµ T i ψL,R ,. (pseudoscalar). (102). (Left, Right),. (103). as above formulae, easy to find the conservation laws Dµ J i,µ =[J 5,i,µ , Aµ ] Dµ J 5,i,µ =[J i,µ , Aµ ] + 2imJpi −DR,µ JRµ =DL,µ JLµ = imJpi ,. (104) (105) (106). for Abelian field, the [J 5,i,µ , Aµ ] and [J i,µ , Aµ ] equal to zero. Following article, we based those conservation laws to do discuss for anomaly Ward-Takahashi identity. The Ward-Takahashi identity is an identity between Green functions that follows from the symmetries of the theory, and that remains valid after renormalization, but where exist a specific case which is anomaly Ward-Takahashi identity.. • Anomaly Ward-Takahashi identity for Abelian field. In this subsection, almost detail from Refs. [66, 67, 99]. 36.

(41) FIG. 14: Lowest-order contributions to Tνµρ of Eq.107.. FIG. 15: Lowest-order contributions to Tµρ of Eq.108. Consider the three-point functions in QED(Abelian field) Z Tνµρ (p1 , p2 , q) = i d4 xd4 y eip1 x+ip2 y · h0|T (Jµ (x)Jρ (y)Jν5 (0))|0i.. (107). and Z Tµρ (p1 , p2 , q) = i. d4 xd4 y eip1 x+ip2 y · h0|T (Jµ (x)Jρ (y)JP (0))|0i.. (108). As the Eq.104 and Eq.105, we have the Vector Ward identity pµ1 Tνµρ = pρ2 Tνµρ = 0. (109). q ν Tνµρ = 2mTµρ. (110). and Axial Ward identity. But when we calculate the lowest-order contributions to Tνµρ and Tµρ , we find that the Ward identity Eq.109 and 110 are not satisfied, Z i i d4 k i Tνµρ = i (−1)tr γµ γν γ5 γρ + (p1 ↔ p2 , µ ↔ ρ), (2π)4 k/ − m k/ + p/1 − m k/ − p/2 − m (111) and Z Tµρ = i. d4 k i i i (−1)tr γµ γ5 γρ + (p1 ↔ p2 , µ ↔ ρ). (112) (2π)4 k/ − m k/ + p/1 − m k/ − p/2 − m 37.

(42) To check the Ward identities, in particular Eq.110, we can use the relation /qγ5 = (p/1 + p/2 )γ5 = γ5 (k/ − p/2 − m) + (k/ + p/1 − m)γ5 + 2mγ5. (113). Multiplied by q ν at Eq.111, and then replace /qγ5 by Eq.113, we have (2) q ν Tνµρ = 2mTµρ + ∆(1) µρ + ∆µρ (1). (114). (2). where ∆µρ and ∆µρ are Z i i i i d4 k (1) ∆µρ = tr γ5 γρ γµ − γ5 γµ γρ (2π)4 k/ + p/1 − m k/ − m k/ − m k/ − p/2 − m. (115). and ∆(2) µρ. Z =. d4 k i i i i tr γ5 γµ γρ − γ5 γρ γµ (2π)4 k/ + p/2 − m k/ − m k/ − m k/ − p/1 − m. (116). (i). If the integrals ∆µρ vanish we have the Ward identity in Eq.110. Superficially this appears to (i). be the case. The two integrals in ∆µρ cancel each other if we can shift the integration variable k to k + q in the second term. But the integrals are linearly divergent and a translation (i). of integration variable produces extra finite terms with ∆µρ 6= 0 . This ruins the Ward identity. The fermion line between the vector and axial vector vertices carries momentum p. We could have chosen to route it differently so that this fermion line carries k + a, where a is some (arbitrary) linear combination of p1 and p2 a = αp1 + (α − β)p2 .. (117). The fact that integral is linearly divergent implies that Tνµρ has an ambiguity in its definition by an amount ∆νµρ(a) =Tνµρ (a) − Tνµρ (0) Z d4 k =(−1) (2π)4 i i i × tr γµ γ5 γρ k/ + a / − m k/ + a / + p/1 − m k/ + a / − p/2 − m i i i − tr γµ γ5 γρ + (p1 ↔ p2 , µ ↔ ρ) k/ − m k/ + p/1 − m k/ − p/2 − m (2) ≡∆(1) νµρ + ∆νµρ . 38. (118) (119) (120) (121) (122).

(43) Applying the result Eq.295 in Appendix, we have Z d4 k τ ∂ i i i (1) ∆νµρ =(−1) tr γµ γν γ5 γρ a (2π)4 ∂kτ k/ − m k/ + p/1 − m k/ − p/2 − m −i2π 2 aτ lim k 2 kτ tr(γµ γα γν γ5 γβ γρ γδ )k α k β k δ /k 6 2 4 k→∞ (2π ) −i2π 2 aδ kδ kα = lim 4iνµρα (2π 2 )4 k→∞ k 2 =δνµρ aδ /8π 2 =. (123) (124) (125) (126). (2). the last line is replacing k δ k α /k 2 by g δα /4. Since ∆νµρ is related to Eq.123 by the exchanges p1 ↔ p2 and µ ↔ ρ, we have (2) ∆νµρ = ∆(1) νµρ + ∆νµρ =. β δνµρ (p1 − p2 )δ . 2 8π. (127). Thus the definition of Tνµρ has an ambiguity signified by the arbitrary parameter β Tνµρ (a) = Tνµρ (0) −. β δνµρ (p1 − p2 )δ ≡ Tνµρ (β). 8π 2. (128). Let us first check the axial Ward identity E.105. Like those in Eq.118 the two surface terms Z tr[(k/ − p/2 + m)γ5 γρ (k/ + m)γµ ] pτ2 ∂ (1) 4 ∆µρ = (129) dk (2π)4 ∂kτ ((k − p2 )2 − m2 )(k 2 − m2 ) pτ2 kτ 2 =− 2iπ lim 2 tr(γα γ5 γρ γβ γµ )k α pβ2 (130) 4 k→∞ k (2π) 1 (131) = − 2 µραβ pα1 pβ2 8π with (1) ∆(2) µρ = ∆µρ .. (132). Thus from Eq.114 and Eq.128, we have q ν Tνµρ (β) = 2mTµρ (0) −. 1−β µραδ pα1 pδ2 . 4π 2. (133). We also consider the vector Ward identity Z d4 k 1 1 1 µ p1 Tµνρ (0) = −1 tr p/1 γν γ5 γρ + p1 · (p1 ↔ p2 , µ ↔ ρ). (2π)4 k/ − m k/ + p/1 − m k/ − p/2 − m (134) 39.

(44) Taken p/1 = (k/ + p/1 − m) − (k/ − m),. (135). thus we can rewrite Eq.134 Z d4 k 1 1 1 1 µ p1 Tµνρ (0) = −1 γν γ5 γρ − γν γ5 γρ tr (2π)4 k/ − m k/ − p/2 − m k/ + p/1 − m k/ − p/2 − m + p1 · (p1 ↔ p2 , µ ↔ ρ). (136) Again the right-hand side is a surface term Z tr[(k/ + p/1 − p/2 + m)γ5 γρ (k/ + p/1 + m)γµ ] pτ1 ∂ µ 4 p1 Tµνρ (0) = dk (2π)4 ∂kτ ((k + p1 − p2 )2 − m2 )((k + p1 )2 − m2 ) pτ kτ = 1 4 2iπ 2 limk→∞ 2 tr(γ5 γν γα γρ γβ )pα2 k β (2π) k 1 α = − 2 ναρβ pbeta 1 p2 8π. (137) (138) (139). the second line is taken k → k − p1 , and the last line is taken Eq.295, sum of results, we have k1µ Tνµρ (β) =. (1 + β) ρναδ pα1 pδ2 . 8π 2. (140). which satisfy the Eq.109 thus we have β = −1, and the axial Ward identity becomes q ν Tνµρ = 2mTµρ −. 1 µραδ pα1 pδ2 . 2π 2. (141). This corresponds to a modification of the axial-vector current divergence Eq.105 as ∂ ν Jν5 (x) = 2iJP (x) +. 1 abcd Fab (x)Fcd (x), (4π)2. (142). where Fab (x) is the usual Abelian gauge boson field tensor. This extra term, the Adler-BellJackiw(ABJ)[66, 67] anomaly, is thus produced by the renormalization effect and has the following properties[99]: (1) The anomaly is Independent of the fermion masses and should also be present in the massless theory. (2) Adler and Bardeen[101] showed that the coefficient in the anomaly term is Not Affected by higher-order radiative corrections, i.e., triangle diagrams with more than one loop don’t contribute to the anomaly term. 40.

(45) (3) It was pointed out by Fujikawa[102] that the ABJ anomalous Ward identity could be formulated rather directly in the path-integral formalism. He showed that the path-integral measure for gauge-invariant fermion theory is Not Invariant under the γ5 transformation. The extra Jacobian factor gives rise to the ABJ anomaly. B.. Anomaly Cancellation. In a (non-abelian) gauge theory, the gauge symmetry is crucial in demonstrating unitarity and renormalizability, and an anomaly in the gauge symmetry would be a disease. The Standard Models is not symmetric under parity, and the W and Z bosons couple directly to chiral currents at the perturbative level. Realized such perturbative anomalies would make the Standard Model non-renormalizable unless the quantum numbers of the fermions were such that all potential pertubative anomalies cancelled. Diagrams that can spoil renormalizablity are shown in Fig.16. For each type, the sum of the two diagrams with the fermions circulating in opposite directions must be added. From the Feynman rules, the group-theoretical factors involving the weak-isopin and strong color matrices, and the hypercharges, can be factored out. The theory will be renormalizable if each such factor vanishes.. FIG. 16: The diagram from Peskin and Schroeder[98]. Possible gauge anomalies of weak interaction theory. All of these anomalies must vanish for the Glashow-Weinberg-Salam theory to be consistent.. For all the diagrams(neglect gravity), in order to ensure the model is renormalizablity we have: 41.

(46) wave function renormalizable request4 U (1)Y –SU (2)L. :. X. :. X. Yf If3 = 0. (143). Yf C f = 0. (144). Yf3 = 0. (145). f. U (1)Y –SU (3)c. f. where I 3 (C) is weak-isospin(colors) element, Anomaly cancellation U (1)Y –U (1)Y –U (1)Y. :. X f. U (1)Y –SU (2)L –SU (2)L. :. X. :. X. (2). (2). (146). (3). (3). (147). Yf tr[Ti Tj ] = 0. f. U (1)Y –SU (3)c –SU (3)c. Yf tr[Ti Tj ] = 0. f (a). where the Tka , a = 2, 3; ka = 1, ..., dimG(a) are the generators of the G(2) = SU (2) and G(3) = SU (3) algebras respectively. In our notation T r[Tia Tka ] = 21 δij and . All the remaining anomalies that involve U (1)0 s vanish identically due to group theoretical arguments (see Chapter 22 of [86] and [57]). The anomaly gauge boson self-energy diagram is nonrenormalizbale, only if that the model added extra scalar to cancel the infinite term(the scalar is doublet Higgs in Standard Model). Taking two examples:. • For U (1)Y − SU (3)c − SU (3)c in the triangle diagram, the overall factor is: X 1 (3) (3) Yf tr[Ti Tj ] = δij (−)L Yquark 2 quark. (148). where the sum runs over quarks with a - sign for the left-handed contributions. Evaluating the sum for the light u and d quarks gives 1 1 2 =0 (−1) × 2 × + + − 6 3 3 Were sequentially are left quark doublet, uR , bR . 4. For SU (2)L − SU (2)L self-energy diagram, the scalar(Higgs) is requirement.. 42. (149).

(47) • For U (1)Y − SU (2)L − SU (2)L in the triangle diagram, the overall factor is: 1 X (2) (2) Yf tr[Ti Tj ] = δij (−)L YfL 2 f. (150). L. Evaluating the sum for all the left hand fermions gives 1 1 + (−1) × 3 × − =0 (−1) × − 2 6. (151). Were sequentially are left lepton doublet, left quark doublet, and the 3 is number of colors.. 43.

(48) V.. MULTILEPTON HIGGS DECAYS THROUGH THE DARK PORTAL. We set up a beyond SM model which include a kinetic mixing between U (1)Y and U (1)D , and the kinetic mixing mechanism will produce non-SM process in multi-leptons final state, that is a topic in this section. In Sec.V A we set up our model. Phenomenology based on similar model has been studied before, see for example Refs. [7–9, 33, 34] and references therein. In Sec.V B we consider mixing effects in the scalar sector as well as the gauge boson sector. We show the hD h mixing in detail and present all the relevant trilinear and quadrilinear couplings of the physical h1 and h2 bosons. We also briefly discuss the mixings between the three neutral gauge bosons in the model as studied previously in Ref. [9]. In Sec.V C we discuss the possible decay modes of the SM Higgs outside those of the SM and their several kinematic regions. In Sec.V D we present numerical results for various branching ratios of the non-standard decay modes of the SM Higgs, identified here as h1 . In Sec.V E we study the signals of multilepton jets of the model at the LHC-14. We conclude in Sec.V F. A.. SU (2)L × U (1)Y × U (1)D Model. We extend the electroweak SM by including the original Abelian Higgs model for a dark U (1)D [7, 8, 33, 34]. The bosonic part of the Lagrangian density is LB = Lgauge + Lscalar. (152). 1~ ~ µν − 1 Bµν B µν − 1 Cµν C µν − Bµν C µν , Lgauge = − W µν · W 4 4 4 2. (153). Lscalar = |Dµ Φ|2 + |Dµ χ|2 − Vscalar (Φ, χ) ,. (154). with. and . 1 1 0 Dµ Φ = ∂µ + ig σa Waµ + i g Bµ Φ , 2 2 Dµ χ = (∂µ + igD Cµ ) χ ,. (155) (156). ~ µ , B µ and C µ are the gauge potentials of the SU (2)L , U (1)Y and U (1)D with gauge where W couplings g, g 0 and gD respectively, and is the kinetic mixing parameter between the two 44.

(49) U (1)s [5]. The scalar potential in (154) is given by Vscalar = −µ2Φ Φ† Φ + λΦ Φ† Φ. 2. − µ2χ χ∗ χ + λχ (χ∗ χ)2 + λΦχ Φ† Φ (χ∗ χ) .. We pick the unitary gauge and expand the scalar fields around the vacuum 1 1 0 , χ(x) = √ (vD + hD (x)) Φ(x) = √ 2 v + h(x) 2. (157). (158). with the VEVs v and vD fixed by minimisation of the potential to be 1 1 1 1 1 4 2 2 − µ2χ vD + λΦχ v 2 vD . Vscalar (v, vD ) = λΦ v 4 − µ2Φ v 2 + λχ vD 4 2 4 2 4. (159). Thus, we obtain 2. v =. µ2Φ −. 1 λΦχ 2 µ 2 λχ χ 2. λΦ −. 1 λΦχ 4 λχ. ,. 2 vD. =. 1 λΦχ 2 µ 2 λΦ Φ 2 λ λχ − 14 λΦχ Φ. µ2χ −. .. (160). In terms of the shifted fields h and hD , the scalar potential Vscalar can then be decomposed as Vscalar = V0 + V1 + V2 + V3 + V4 with 1 4 2 2 λΦ v 4 + λχ vD + λΦχ v 2 vD − 2µ2Φ v 2 − 2µ2χ vD , (161) 4 1 1 2 2 = v 2λΦ v 2 + λΦχ vD − 2µ2Φ h + vD 2λχ vD + λΦχ v 2 − 2µ2χ hD , (162) 2 2 3 3 1 1 1 1 2 2 λΦ v 2 + λΦχ vD − µ2Φ h2 + λχ vD + λΦχ v 2 − µ2χ h2D + λΦχ vvD hhD , = 2 4 2 2 4 2 1 , (163) ≡ (h hD ) · MS2 · h 2 hD 1 = λΦ vh3 + λχ vD h3D + λΦχ vD hD h2 + vhh2D , (164) 2 1 1 1 (165) = λΦ h4 + λχ h4D + λΦχ h2 h2D . 4 4 4. V0 = V1 V2. V3 V4. Here V0 is a cosmological constant and will be discarded from now on; the tadpole term V1 vanishes with v and vD given by Eq.(160); V2 is quadratic in the fields h and hD , and we have to diagonalize the mass matrix MS2 in Eq.(163) to get the physical Higgs fields h1 and h2 (see next section); and V3 and V4 are the trilinear and quadrilinear self couplings among the two Higgs fields. Since χ is a SM singlet, the W and Z bosons acquire their masses through the SM Higgs doublet VEV v entirely which implies v ∼ 246 GeV. 45.

(50) B. 1.. Mixing Effects Higgs Mass Eigenstates and Their Self Interactions. The mass matrix MS2 in Eq.(163) for the scalar bosons is 2 2 m m 2 12 11 , MS = m221 m222 2λΦ v 2 λΦχ vvD = . 2 λΦχ vvD 2λχ vD. (166). Its eigenvalues are m21,2. q 1 2 2 = TrMS ± (TrMS2 ) − 4 DetMS2 . 2. The physical Higgs (h1 , h2 ) are related to the original (h, hD ) as h1 = cos α sin α h , h2 − sin α cos α hD. (167). (168). with the mixing angle sin 2α =. 2m212 . m21 − m22. (169). We will identify the heavier Higgs h1 with mass m1 = 126 GeV as the new boson observed at the LHC [35, 36], while the lighter one h2 has been escaped detection thus far. In terms of the physical Higgs fields h1 and h2 , the cubic term V3 is given by V3 =. 1 (1) 3 1 (2) 3 1 (3) 1 (4) λ3 h1 + λ3 h2 + λ3 h1 h22 + λ3 h2 h21 3! 3! 2 2. with the trilinear couplings 1 (1) 3 3 λ3 = 3 2vλΦ cos α + 2vD λχ sin α + λΦχ sin 2α (v sin α + vD cos α) , 2 1 (2) 3 3 λ3 = 3 −2vλΦ sin α + 2vD λχ cos α + λΦχ sin 2α (vD sin α − v cos α) , 2 1 (3) λ3 = 24vλΦ sin2 α cos α + 24vD λχ cos2 α sin α 4 +λΦχ (v cos α + vD sin α + 3v cos 3α − 3vD sin 3α)] , 1 (4) λ3 = −24vλΦ sin α cos2 α + 24vD λχ cos α sin2 α 4 +λΦχ (−v sin α + vD cos α + 3v sin 3α + 3vD cos 3α)] , 46. (170). (171) (172). (173). (174).

(51) and the quartic term V4 is given by V4 =. 1 (1) 4 1 (2) 4 1 (3) 1 (4) 1 (5) λ4 h1 + λ4 h2 + λ4 h1 h32 + λ4 h2 h31 + λ4 h21 h22 4! 4! 3! 3! 2! · 2!. with the quadrilinear couplings (1) λ4 = 6 λΦ cos4 α + λχ sin4 α + (2) λ4 = 6 λΦ sin4 α + λχ cos4 α +. 1 2 λΦχ sin 2α , 4 1 2 λΦχ sin 2α , 4. 3 (3) λ4 = − sin 2α −2λχ cos2 α + 2λΦ sin2 α + λΦχ cos 2α , 2 3 (4) λ4 = + sin 2α 2λχ sin2 α − 2λΦ cos2 α + λΦχ cos 2α , 2 1 (5) λ4 = [3 (λΦ + λχ ) + λΦχ − 3 (λΦ + λχ − λΦχ ) cos 4α] . 4 2.. (175). (176) (177) (178) (179) (180). Kinetic and Mass Mixing of the neutral gauge bosons. In additional to the mass mixing of the three neutral gauge bosons arise from the spontaneously electroweak symmetry breaking given by . Cµ. . 1 Lm = (C µ B µ W 3µ ) · M 2 · Bµ 2 Wµ3. (181). with the following mass mixing matrix . 2 2 gD vD 2 M = 0 0. 0 1 2 2 g v 4 Y 1 − 4 ggY v 2. 0 − 41 ggY v 2 . 1 2 2 g v 4. (182). we also have the kinetic mixing between the two U(1)s from the last term in Eq.153. Both the kinetic and mass mixings can be diagonalized simultaneously by the following mixed transformation [76] . Cµ. . . AD,µ. . Bµ = K · O Zµ , Wµ3 Aµ 47. (183).

(52) where AD, , Z and A are the physical dark photon, Z boson and the photon respectively. Here K is a general linear transformation that diagonalizes the kinetic mixing β 00 K = −β 1 0 , 0 01 where β =. √ 1 1−2. (184). with < 1, and O is a 3 × 3 orthogonal matrix which can be parametrized. as . cos ψ cos φ − sin θ sin φ sin ψ cos ψ sin φ + sin θ cos φ sin ψ − cos θ sin ψ. sin ψ cos φ + sin θ sin φ cos ψ sin ψ sin φ − sin θ cos φ cos ψ cos θ cos ψ. − cos θ sin φ cos θ cos φ sin θ. (185). with the mixing angles defined as gY , tan θ = g. − tan φ = √ , 1 − 2. 2 tan φ cos θ 1 − m2z /MW 2 + tan θ . (186) tan ψ = ± 2 2 tan θ 1 − m2Z /gD vD. After the K transformation, the gauge bosons mass matrix is 2 2 2 β (gD vD + 41 2 gY2 v 2 ) − 41 βgY2 v 2 14 βggY v 2 1 2 2 ˜ 2 = KT · M 2 · K = M − 41 ggY v 2 . g v − 14 βgY2 v 2 4 Y 1 1 2 2 − 41 βggY v 2 ggY v 2 g v 4 4. (187). ˜ 2 matrix The O matrix diagonalize this M m2γD 0 0 ˜ 2 · O = 0 m2 0 , = OT · M z 0 0 m2γ . 2 MDiag. (188). with the following eigenvalues (assuming mγD ≤ mZ 5 ) m2γ = 0,. m2z,γD = (q ± p)/2. (189). where q 2 2 2 2 p = q 2 − gD vD v (g + gY2 )β 2 ,. (190). 1 2 2 2 q = gD vD β + (g 2 + gY2 β 2 )v 2 . (191) 4 For small kinetic parameter mixing , the Z and γD masses can be approximated by p mz ≈ (g 2 + gY2 )v/2 and mγD ≈ gD vD .. 5. For the case of mγD > mz , we will have m2z,γD = (q ∓ p)/2 the case which has been studied previously [76], which diagonal method is different with [33, 34]. 48.

(53) C.. Non-standard Decays of h1. The global fits [9, 11, 12, 60] for the signal strengths of the various SM Higgs decay channels from the LHC data imply the total width of the SM Higgs is about 4.03 MeV and the non-standard width for the SM Higgs can be at most 1.2 MeV; in other words the non-standard branching ratio for the SM Higgs must be less than 22%. One can use this result to constrain the parameter space of the model. We will compute the following non-standard processes h1 → γD γD , h1 → h2 h2 , h1 → h2 h∗2 → h2 γD γD and h1 → h2 h2 h2 . Each of the h2 in the final state of these processes will decay into two dark photons and each dark photon will give rise to two leptons through its mixing with the photon 6 . These non-standard processes will provide multiple leptons in the final state of the standard model Higgs decay [7]. The contribution to the heavier Higgs width from these non-standard processes is. 7. S 2 ˆ ΓN h1 = sin αΓ(h1 → γD γD )+Γ(h1 → h2 h2 )+Γ(h1 → h2 γD γD )+Γ(h1 → h2 h2 h2 )+· · · (192). Thus the total width of the heavier Higgs h1 is modified as S ˆ h + ΓN Γh1 = cos2 αΓ h1 ,. (193). ˆ h is the width of the SM Higgs h, which has a theoretical value of 4.03 MeV. The where Γ branching ratio for the non-standard modes of the heavier Higgs decay is BhN1S. S ΓN h1 = , Γh1. (194). which should be constrained to be less than 22% or so. The partial decay width for the two body decays are given by 2 gD m2γD ˆ Γ(h1 → γD γD ) = 8πm1. 1 4m2γD 2 m41 m21 1− 3− 2 + , m21 mγD 4m4γD. and . (3). 2. 1 4m22 2 Γ (h1 → h2 h2 ) = 1− 2 . 32πm1 m1 6 7. λ3. (195). (196). We note that h2 can decay to SM particles as well through its mixing with h1 and hence they are suppressed. We take the branching ratio of h2 → γD γD to be 100%. See discussion after Eq.(202). The four lepton modes from the first term h1 → γD γD followed by γD → l¯l (l = e, µ) were studied in details in [33].. 49.