Effective description of the EW symmetry breaking

Transcription

1 Author:. Facultat de Física, Uniersitat de Barcelona, Diagonal 645, 0808 Barcelona, Spain. Adisor: Dr. Domènec Espriu The discoery of the Higgs boson in 01 was a huge moment of achieement: the particle postulated more than 50 years ago was at last discoered. Een so, the particle acted as it was expected and, paradoxically, gae no new clues about to where to look next. In this report we briefly discuss what we call the Higgs mechanism, which gies mass to the W and Z when they interact with an inisible field, the Higgs field, that perades the unierse. Moreoer, the Electroweak Chiral Lagrangian is presented, which allows us to consider an alternatie way to perform an analysis of the model. A brief outline and some final reflections are exposed in the conclusion section. I. INTRODUCTION One of the greatest achieements in the history of physics is the deelopment of a theory such as the Standard Model of fundamental interactions (SM). Three of the four known fundamental forces in the unierse (the electromagnetic, weak and strong interactions) and how the basic building blocks of matter interact are ery accurately explained by this well-tested physics theory. Since the 0th century, principles of symmetry hae been playing a critical role in fundamental physics, especially in quantum field theory. Moreoer, an appealing connection is set between the gauge symmetries and the fundamental interactions. In a more formal manner, it is well established that the theoretical framework of the SM is based on the gauge symmetry group SU(3) C SU() L U(Y) Y. While the SU(3) C gauge group describes the strong interaction, the SU() L U(Y) Y is connected to the electroweak interaction. Een though the SM seemed at first sight to be a complete description of the subatomic world, it was found experimentally that the mediators of the electroweak interaction, the W +, the W and the Z, had masses different from zero. At that time, it was well understood that this fact was strictly forbidden by the gauge symmetry in the SM. Because of this, and because of others aspects, the so-called electroweak symmetry SU() L U(Y) Y had to be necessarily broken. On 4 July 01, the ATLAS and CMS experiments at CERN s Large Hadron Collider (LHC) reealed one of the models proposed to explain the aforementioned symmetry breaking to be the correct one: the Higgs model. In fact, what the experiments at CERN s discoered was a new particle in the mass region around 16 GeV consistent with the Higgs Boson, the transparent manifestation of the Higgs mechanism. Consequently, this led to the award of the Nobel prize in physics to François Englert and Peter Higgs on 8 October 013 [1]. The aim of the present work, therefore, is to analyse Electronic address: oarandte7@alumnes.ub.edu some general effectie theory that enables us to comprehend the Higgs mechanism with a different approach. The report is organized as follows. Section II introduce the idea of Effectie Field Theories and Effectie Lagragians, a general tool to describe Field Theories only constructed by symmetry considerations. Goldstone s Theorem is briefly mentioned in Section III to bring forward the Goldstone bosons, bosons that arise when a continuous symmetry is broken. Their introduction is determined by the critical role they will play in further deelopment. In Section IV we find a major section that explores the so-called Higgs mechanism in two different manners, the latter being of extreme releance to reeal an important symmetry hidden in the releant Lagrangian. And finally, the paper ends in Section V by showing the construction of a general Lagrangian with all important symmetries that allows us to see beyond the Standard Model Lagrangian terms. II. EFFECTIVE FIELD THEORIES The theoretical framework known as the Standard Model (SM) is a quantum field theory that describes the electroweak and strong interactions of quark and leptons. Schematically, it consists of: A matter sector which is made of fermionic fields. Vector boson gauge fields. A symmetry breaking sector (SBS). The last one is needed to proide masses for the fermions and EW bosons and it is not as well established as the other two. In the aforementioned symmetry breaking sector we find the issue under discussion: the electroweak symmetry breaking (EWSB). A ery appealing option to describe this symmetry breaking is to use effectie theories. An Effectie Field Theory is a quantum field theory that enables us to study the releant physics of our system without the need of specifying a particular model at high energies. It can be constructed from the releant symmetries of the light modes at low energies. Nonetheless,

2 both light and heay degrees of freedom are included, the latter haing been integrated out. One of the simplest theory that allows us to describe the EWSB is the Electroweak Chiral Lagrangian (ECL). The choice of the name for the ECL is due to the similarity between the EWSB pattern and the one of the chiral symmetry in Quantum Chromodyamics. III. THE GOLDSTONE THEOREM Models exhibiting spontaneous breakdown of continuous symmetries lead to the appearance of massless particles. This is a general result known as Goldstone theorem. For each generator of the broken symmetry we hae these massless particles known as Goldstone bosons (GB). In the case of Quantum Chromodynamics (QCD) we hae an approximate symmetry in L QCD known as the chiral symmetry. When it is spontaneously broken, the Goldstone bosons associated to his breaking are identified (at least approximately) with the three pions, π ± and π 0. This example is one of the many light bosons seen in physics that may be interpreted as Goldstone bosons. In the Electroweak Chiral Lagrangian the EWSB gies three Goldstone bosons, w +, w, and w 0, corresponding to the longitudinal degrees of freedom of the electroweak mediators, W +, W and Z, respectiely. Moreoer, the Higgs boson could also be interpreted as en extra Goldstone boson in some models such as the Minimal Composite Higgs Model [], but this is not necessarily so. A ery transparent but general proof of Goldstone s theorem for classical scalar field theories is presented in [3]. The idea is to apply the concept of spontaneous symmetry breaking to the SU() U(1) model of the electroweak interactions. The most general SU() L U(1) Y inariant potential depends only on the combination Φ Φ Φ. Here we follow the particular form of the potential presented in [4]: V (Φ) = λ(φ + µ λ ) (4) with λ and µ arbitrary parameters. Furthermore, the literature commonly adopts an equialent potential V (Φ) = µ Φ Φ + λ(φ Φ), where the constant factors are remoed and the λ and µ parameters are properly redefined [5][3]. Minimizing the potential V we find two solutions, the triial solution Φ 0 = 0 and the nontriial solution: Φ Φ 0 = µ λ (5) Therefore, we hae a minimum away from the origin proided that the quantity µ in the potential is negatie. IV. THE HIGGS MECHANISM A. A first approach The simplest way to gie masses to the W s and Z is ia the complex doublet of spin-zero Higgs field: ( ) φ + Φ = φ 0 = 1 ( ) φ1 iφ (1) φ 3 + iφ 4 FIG. 1: Potential V(Φ) as a function of Φ for the positie sign of the µ term. Triial solution for the acuum implies that the alue of all the fields φ i in the minimum energy state is zero. where the superscripts are the Q electric charge assigments according to Q = Y + T 3. The conentional ensures that the fields are normalized in the same way. The lagrangian density that describes this scalar sector reads as: L = (D µ Φ) (D µ Φ) V (Φ Φ) () where the coariant deriate is defined as: D µ Φ = ( µ + ig W µ T + ig B µ Y )Φ = ( µ + ig W µ τ + ig B µi)φ (3) FIG. : Potential V(Φ) as a function of Φ for the negatie sign of the µ term. Nontriial solution for the acuum implies that at least one of the four fields (φ i ) must be non-zero. Hence, in general, we cannot treat all four components of Φ in a symmetric manner. Treball de Fi de Grau Barcelona, June 017

3 A nontriial accum Higgs configuration which obeys the constraint Eq.5 and presere the quantum properties and quantum numbers of the acuum is: Φ 0 = 1 ( 0 ) (6) for real. Of course, this is purely conentional: one can make an SU() U(1) transformation to make any accum expectation alue (VEV) of Φ and Φ 0 hae the aforementioned form. See a demonstration in [4]. B. Spontaneous symmetry breaking It is conenient now to introduce the following conjugate doublet: And introduce the matrix [6] Φ iτ Φ (7) M(x) = ) ( ΦΦ = ( φ 0 ) φ + φ φ 0 (8) which allows us to recast the Standard Model Lagrangian inoling the Higgs doublet in a completely equialent form as: with and L SBS = 1 4 T r[(d µm) (D µ M)] 1 4 λ[1 T r(m M) + µ λ ] (9) D µ M = µ M + L µ M MR µ (10) L µ M = ig τ W µ R µ = ig τ 3 B µ (11) It is easy to show now that the Lagrangian L SBS is inariant under the so-called electroweak chiral symmetry SU() L SU() R. One can check it by performing two global and independent chiral transformations on M M LMR L, R SU() L,R (1) and check that the L SBS remains the same. That symmetry was not so obious before introducing the equialent formalism with matrix M. Howeer, this hidden symmetry now appears as eident. The interesting issue is that this symmetry is spontaneously broken into the custodial symmetry group SU() L SU() R SU() C = SU() L+R (13) To see that, it is useful for our purpose to notice that the matrix M can be written as M(x) = σ(x)u(x), where σ(x) is a real scalar field and U(x) is an SU() field. Computing M M one can get coninced that σ = ( φ 0 + φ + φ ) 0. Thus, defining the quantity µ λ to be positie the Lagrangian reads: L = 1 µσ µ σ + σ 4 T r[(d µu) (D µ U)] λ 4 (σ µ λ ) (14) If we minimize the potential as we did before, we get, apart from the triial solution σ = 0 (which corresponds to unestable maximum), the nontriial accum expectation alue (VEV): µ σ = ± ± (15) λ which implies: M 0 = ( ) 0 = I (16) 0 Therefore, the unitary matrix U in the acuum turns out to be U 0 = I and now the acuum is left inariant only if L= R. Hence, the global symmetry group SU() L SU() R has been broken to SU() L+R and the three Goldstone bosons associated to symmetry breaking are contained in the matrix U. Ignoring the fluctuations around the acuum, we replace M U to get L = 4 T r(d µud µ U ) (17) To interpret this theory, suppose that the system is near one of the minima (say the positie one). Then it is conenient to define the shift field σ = σ (18) and rewrite L in terms of σ. Dropping the constant term, equation 14 becomes: L = 1 µ σ µ σ + = 1 µ σ µ σ + ( σ + ) T r[(d µ U) (D µ U)] 4 λ 4 (( σ + ) µ λ ) ( σ + ) T r[(d µ U) (D µ U)] 4 1 ( µ) σ λ σ 3 λ 4 σ4 (19) This Lagrangian describes a simple scalar field of mass M H = µ = λ with σ 3 and σ 4 interactions. This is the so-called Higgs field, whose mass is usually represented by M H. Predicted more than 50 years ago, the Higgs boson was at last discoered with a mass nowadays of M H = ± 0.1 GeV [7]. Treball de Fi de Grau 3 Barcelona, June 017

4 V. ELECTROWEAK CHIRAL LAGRANGIAN In the Electroweak Chiral Lagrangian (ECL), the Electroweak Goldstone bosons (EW GB) are introduced in a non-linear exponential representation U = exp (i σa a ), (0) where a are the EW GB fields, σ a are the Pauli matrices, both for a=1,,3 and = 46GeV is the VEV of the SM scalar doublet. Our aim is to construct the most general ECL of QED, with interactions terms inariant under chiral SU() transformations. The simplest chiral and Lorentz inariant term inoling the U field is the deriatie-free term [5]: T r(uu ) = (1) Since this is a constant it can always be remoed from the Lagrangian. Thus, the general Lagrangian will contain terms with arbitrary number of deriaties classifiying the operators according to their energy dimension: L = L + L 4 + L () where L n denotes generally the term with n deriaties. Therefore, we are able to construct the operators of the ECL order by order compatible with the releant symmetries. In a general sense, we consider an analytic function of σ that is locally gien by: The Lagrangian inoling terms with four deriaties depends on but also on other effectie coefficients usually known as chiral parameters or chiral coefficents. They are of great importance since they encode the information on the heay modes that hae been integrated out. As they parametrize the interaction between EW gauge bosons and the Higgs, the determination of their numerical alue will proide us critical information about the mechanism under the EWSB. But, moreoer, an appeling issue is that different sets of alues for chiral parameters correspond to different theories, including the SM. That is, potential desiations from the SM predictions, that could be detected experimentally, depend on the alue of theses couplings or parameters. A. Tree Leel We are able now to extract the Tree Leel interactions of these two contributions to the Lagrangian, L and L 4. This can be done by expanding the exponential as U = 1+ iσa a σa σ b a b iσa σ b σ c a b c (7) For the sake of simplicity, we will omit the B µ field in the calculation due to no conceptual change is made in this. For our purpose, we will only keep those terms inoling up to four fields or two fields with one W a µ. f(σ) = f(0) + f (0)σ + 1 f (0)σ +... (3) This way, at the lowest order of the Higgless EW Chiral Lagrangian we will hae: L = 1 4 [f (σ)] T r(d µ UD µ U ) = 1 4 [ + aσ + (b + a )σ +...]T r(d µ UD µ U ) (4) where we hae defined f (0), a f (0), b f (0) and so on. For the next-to-leading order Lagrangian L 4 = (a )[T r((d µ U)U (D ν U)U )] +(a )[T r((d µ U)U (D µ U)U )] (5) with a 4, a 5 f 4 (0) and always understanding the coariant deriatie of the matrix U as: D µ U = µ U + i g W a µ σ a U i g UB µσ 3 (6) with the Pauli matrices σ a and a the EW GB fields (a=1,,3). Hence, we hae constructed a more general Lagrangian than the one gien by the Standard Model. Indeed, replacing a = b = 0 and a 4 = a 5 = 0 we recoer the Lagrangian gien by equation 17. FIG. 3: Diagrams contributing at tree leel The terms that appear in the Higgless EW Chiral Lagrangian are presented below: L = 1 µ a µ a [(a µ a )( b µ b ) W ( b µ a )( b µ a )] gɛabc µ a W bµ c (8) L 4 = 4 a 4 4 µ a ν a µ b ν b +4 a (9) 5 4 µ a µ a ν b ν b where we hae used the well-known trace properties of Pauli matrices tr(σ a σ b ) = δ ab tr(σ a σ b σ c ) = iɛ abc tr(σ a σ b σ c σ d ) = (δ ab δ cd δ ac δ bd + δ ad δ bc ) (30) Notice there is no mass term for the aboe Lagrangians since they describe Goldstone bosons. Treball de Fi de Grau 4 Barcelona, June 017

5 B. Feynman Rules In pursuit of a more detailed analysis, we are able to compute the Feynman rules from the interaction terms of L and L 4 Lagrangians. For the first interaction term in L Lagrangian we find: i 3 [(δab δ cd + δ ad δ bc δ ac δ bd )t +(δ ab δ cd + δ ac δ bd δ ad δ bc )u (δ ad δ bc + δ ac δ bd + δ ab δ cd )s] where we hae introduced the Mandelstam ariables: s = (p 1 + p ) = (p 3 + p 4 ) (31) t = (p 1 p 3 ) = (p p 4 ) u = (p 1 p 4 ) = (p p 3 ) (3) Likewise, the second term gies the following contribution gɛ abc (δad δ ce p 1µ + δ ae δ cd p µ )W bµ (33) For the L 4 contribution we put Feynman rules for the first term as an example. VI. ia 4 4 [(3δab δ cd + 4δ ad δ bc )t +(4δ ab δ cd + 4δ ac δ bd )u +(δ ab δ cd + 4δ ac δ bd + 4δ ad δ bc )s ] OUTLOOK AND CONCLUSIONS (34) Throughout this work we hae dealt with electroweak chiral symmetry SU() L SU() R, a global symmetry included in the electroweak Standard Model and with the required global symmetry breaking pattern SU() L SU() R SU() L+R. After presenting a first approach of the Higgs mechanism, an equialent formalism assisted by the aboementioned matrix M is deeloped. It is worth highlighting that the Higgs field is introduced ad hoc in the theory. That is, releant quantities of the Higgs such as the Higgs self coupling cannot be predicted. Therefore, there are some issue left that SM cannot explain yet. The fact that matrix M could always be written as M = σu was of great help for a better understanding. One can appreciate that the potential in equation 14 is only a function of σ, V=V(σ). Thus, the components of M that are not σ (i.e. U) lead us to a different but equialent acua. Indeed, moing along these directions cost no energy at all. In the final section we focus in a ery appeling way of approaching to the matter under discussion. Effectie Lagrangians brings us the opportunity to consider a more general iew of this kind of theoretical descriptions. Based on the releant symmetries, we are able to treat with more interactions terms beyond those ones predicted by the SM. These terms could change the strengh of the interactions or bring some new effects that could, eentually, be detected experimentally. Acknowledgments I would like to sincerely thank my adisor Dr. Domènec Espriu for his patience and guidance during the deelopment of this paper. I want to express him all my gratitude for introducing me into some features of the SM. I also want to thank my parents and sister for the constant support and confidence they hae placed on me all this time. [1] The Nobel Prize in Physics 013. Retrieed from [ [] K. Agashe, R. Contino, A. Pomarol, Nucl. Phys. B719, (005), arxi:hep-ph/ [hep-ph]. [3] M.E. Peskin, D.V. Schroeder, An Introduction to quantum field theory, Collæge Press. Uniersity of Beijing, [4] H. Georgi, Weak interactions and modern particle theory, Doer Publications, 009. [5] J.F. Donoghue, E. Golowich, and B.R. Holstein, Dynamics of the standard model, Cambridge Uniersity Press, 199. [6] A. Dobado, A. Gomez-Nicola, A.L. Maroto, and J.R. Pelaez, Effectie lagrangians for the standard model, Springer, [7] G. Aad et al. (ATLAS, CMS), Phys. Re. Lett. 114, (015), arxi: [hep-ex]. Treball de Fi de Grau 5 Barcelona, June 017