QED and the Standard Model Autumn 2014

Transcription

1 QED and the Standard Model Autumn 2014 Joel Goldstein University of Bristol These lectures are designed to give an introduction to the gauge theories of the standard model of particle physics (SM). No formal quantum field theory is assumed so there are times when rigour will be lacking. In the limited time available, the goal will be to understand the underlying physics of a single generation of fermions and their interactions via strong, electromagnetic and weak interactions. Flavour physics, CP violation and many other interesting topics will therefore have to be saved for later courses. Contents 1 Lagrangian Mechanics 3 2 Noether s Theorem 4 3 Gauge Invariance and QED 5 4 QED Calculations 6 5 Non-Abelian Gauge Transformations 7 6 QCD 8 7 The Weak Force 9 8 Spontaneous Symmetry Breaking 10 1

2 9 Fermion Masses Gauge Boson Masses Electroweak Interactions Standard Model Summary Loops and Renormalisation 14 Introduction The SM encapsulates our current understanding of the fundamental particles of nature and their interactions. It has survived rigorous testing over the past 40 years or so with only minor modifications (e.g. the addition of neutrino masses), although it is known to be incomplete. Mathematically the SM is a quantum field theory, and although a complete grasp of this formalism is not necessary to understand much of the physics it should be kept in mind that the objects the theory describes are quantum field operators. Such field theories are traditionally approached through Lagrangian mechanics, which we will briefly review. One of the most pleasing aspects of the SM is that almost all of the physics comes directly from the symmetries that are imposed. In particular, the interactions are produced by requiring local gauge invariance under certain symmetry groups. We will therefore spend some time examining the effect of different symmetry transformations on our physics. The gauge symmetry groups of the SM are U(1) SU(2) SU(3) and the fermion fields of the first generation are (u, d) and (ν, e). These are chosen specifically to try to produce the universe we live in, but this attempt fails spectacularly. In order to save it, we will introduce ad-hoc a new scalar field with a non-zero vacuum expectation value which spontaneously breaks the symmetry of our underlying model via the Higgs mechanism. Note that we will use Natural Units throughout in which all factors of h and c are implied rather than written. 2

3 1 Lagrangian Mechanics In classical mechanics, the equations of motion of a system can be derived from Hamilton s Principle. This states that the path the system follows (in configuration or phase space, a 2N-dimensional space in which the coordinates x i and their time derivatives ẋ i are independent variables) is the path s which minimises the action S where S = Lds and the Lagrangian L is taken to be the difference between the kinetic and potential energy: L = T V Hamilton s Principle leads directly to the Euler-Lagrange equation(s) of motion: d L = L dt q i q i This formalism can be extended to relativistic fields by introducing the Lagrangian Density L(φ, µ φ) such that L = d 4 xl The Euler-Lagrange equation for the field φ then becomes µ δl δ( µ φ) = δl δφ (1) where the s have formally become δs to indicate that we are taking the derivatives with respect to functions rather than coordinates. [Note - I will revert to using even for functional derivatives in future, to avoid confusion with δ representing a small variation]. Particle physics is largely a search for the Lagrangian that gives the desired equations of motion. Consider for example the Lagrangian density L = 1 2 ( µφ)( µ φ) 1 2 m2 φ 2 (2) Applying Equation 1 gives the equation of motion ( µ µ + m 2 )φ = 0 3

4 which is the Klein-Gordon equation. Equation 2 is thus the Lagrangian density for a free, real, spin-0 field. The first term gives the kinematics of the field and the second gives the mass. Similarly, a Lagrangian density L = iψ/ ψ mψψ (3) describes fermionic fields, leading to the Dirac Equation for ψ and ψ. Defining the electromagnetic field strength tensor in terms of the vector potential A µ F µν = µ A ν ν A µ the Lagrangian density L = 1 4 F µνf µν (4) unsurprisingly leads to Maxwell s Equations: ν F µν = 0 2 Noether s Theorem A transformation that explicitly leaves the Lagrangian (and hence the equations of motion) unchanged is a symmetry transformation. For our purposes, we will consider the effect of a continuous symmetry transformation of the fields only, i.e. for an infinitesimal change φ i φ i = φ i + δφ = φ i + Φ ij ɛ j (Note that this general form allows for transformations between fields for i j). Requiring that this leaves the Lagrangian invariant leads to the equation µ J µ i = 0 (5) with J µ i = L ( µ φ j ) Φ ji which predicts the existence of i = 1...n conserved currents. Each current has an associated conserved charge obtained by integrating Ji o at a particular time Q i = d 3 xji 0 4

5 This is an example of Noether s Theorem which predicts that any continuous symmetry will lead to a conserved current and charge for each parameter. A similar treatment for coordinate transforms only (i.e. x µ x µ + X µν x ν ) gives conservation of energy and momentum, both linear and angular. 3 Gauge Invariance and QED The invariance of a system under global transformations in the potential, V V +, is very familiar. More generally, we can change the vector potential by adding any function that is a simple derivative: A µ A µ = A µ µ ξ (6) In quantum mechanics, we can instead start by considering changes to the phase of the wavefunctions or fields. For example, the Dirac Lagrangian in Equation 3 is clearly invariant under phase rotations ψ e iqθ ψ, ψ e iqθ ψ, and thus we have a U(1) symmetry and corresponding conserved current. The infinitesimal form ψ ψ + iq(δθ)ψ shows that Φ ij in Equation 5 is a single term (as we expect since there is only one transformation parameter) and is equal to iqψ. The conserved current from Equation 5 is then J µ = qψγ µ ψ which is exactly what we would expect for the electromagnetic current of fermions with charge q. We can now go a step further and see what happens if we require that the gauge transformation is a function of position, i.e. ψ e iqθ(x) ψ. This is known as a local (as opposed to global) transformation. For an infinitesimal transformation e iqθ(x) = 1 + iqθ(x) and (ignoring terms to second order in θ) L L qψ( θ)ψ so the Lagrangian is no longer invariant. Instead of giving up on local gauge invariance, we hypothesise that our fermions interact with a gauge field A µ. The interacting fermion Lagrangian will be L = ψ(iγ µ D µ m)ψ (7) where we have defined the covariant derivative D µ = µ + iqa µ 5

6 and we also insist that A µ undergoes the gauge transformation A µ A µ µ θ(x) as in Equation 6. The effect is that the gauge transformation of the gauge field cancels that of the fermion fields and invariance is restored. The Lagrangian in Equation 7 is exactly what we would expect for a fermion of charge q in an electromagnetic field. The qψγ µ A µ ψ term represents the interaction of a fermion and a vector field with coupling strength q = Qe where Q is the charge of a particle in units of e, the coupling constant. By insisting on local U(1) gauge invariance we have forced ourselves to invent QED. The only thing left is to put in the gauge field (photon) kinematic term from Equation 4 giving the complete QED Lagrangian: L = ψ(i /D m)ψ 1 4 F µνf µν (8) Note that a gauge field mass term, M 2 A µ A µ would explicitly break the gauge invariance. 4 QED Calculations Perturbation theory gives the transition rate W fi from state i to state f as W fi = M fi 2 (2π) 4 δ 4 (p f p i ) where the delta function ensures energy and momentum conservation. The transition amplitude (or matrix element) due to an interaction potential V int is M fi = f V int i = f L int i. This can be used to calculate decay rates and cross sections by integrating over the final state. Two particularly useful results are the formula for the width of the decay of a particle of mass m into two particles with equal and opposite 3-momentum ± p: Γ = 1 32π 2 m 2 M 2 p dω (9) and the formula for the differential cross section of the scattering process a + b c + d in the centre-of-mass frame: dσ dω = 1 64π 2 s 6 p c p a M 2 (10)

7 where s is the usual Mandelstam variable s = (p a + p b ) 2 = (p c + p d ) 2. The matrix element f L int i is most easily calculated from the Feynman rules. Multiplying the contribution from each component gives im at tree level: External fermion states f and i give the particle spinors u f and u i, and the antiparticle spinors v f and v i. External photon states f and i give the polarisation vectors ɛ f and ɛ i. Internal particle lines must satisfy the relevant free particle equations of motion, giving the propagators ig p 2 + iɛ i( p + m) p 2 m 2 for photons for fermions The coupling to the interaction potential gives the vertex term ieγ 5 Non-Abelian Gauge Transformations Not all gauge transformations commute as the U(1) of QED does. Consider a gauge transformation between a multiplet of n fields: φ i Φ ij φ j The matrix Φ is a member of the group SU(n), which is non-abelian. Since there are n 2 1 degrees of freedom in an arbitrary SU(n) matrix, in infinitesimal form it can be written as Φ = 1 igω k T k where the n 2 1 T matrices are known as the generators of the group. Requiring invariance under such a local SU(n) gauge symmetry will (in similarity to above) lead us to introduce a covariant derivative D µ = I µ iga µ (11) 7

8 where the gauge fields are A µ = T k A k µ and have a field-strength F µν = T k F k µν. Unfortunately, since the fields do not commute we find that where f klm is a structure constant. F k µν = µ A k ν ν A k µ gf klm A l µa m ν The Lagrangian for a fermionic field with a local SU(n) gauge symmetry is then L = ψ(i /D m)ψ 1 4 F k µνf k µν (12) As well as the fermion kinematic term this contains the coupling of the fermion to n 2 1 vector gauge fields with strength g and the corresponding gauge boson kinematic terms. Notice that gauge boson mass terms are still forbidden by gauge invariance, but that the non-zero structure constants now give terms of the form ga k A l A m and g 2 A l A m A n A o, i.e. three- and four-point interactions between gauge bosons. 6 QCD Experimental evidence (e.g. measurement of R) is that quarks carry a conserved charge called color with three degrees of freedom, so a quark can be represented as a multiplet of fields in color space: c = If we assume that there is a local symmetry for transformations in color space we must have an SU(3) gauge symmetry, and hence 8 gauge fields, the gluons. These correspond to the 8 SU(3) generators, normally written as the Gell- Man matrices λ As described in the previous section, the gluons have self interactions which leads to the fundamentally different phenomenology of QCD and QED. Notice that all leptons must be singlets under this SU(3) symmetry as they do not couple to the gluons. The Feynman rules for QCD are very similar to those in QED as described in Section 4, with the addition of SU(3) charges for the gluon and quarks. (There are also three and four-point gluon self-interaction vertices which we will not discuss further). The SU(3) terms are: the SU(3) charge g s /2 replaces the QED charge q. 8 r g b

9 a three-component color wavefunction c for an incoming quark or outgoing antiquark an adjoint color wavefunction c + = (r, g, b) for an outgoing quark or incoming antiquark a Gell-Man matrix λ a for each quark-gluon vertex a Kronecker delta δ aa for each quark to ensure color conservation Since the SU(3) terms commute with spinors and γ matrices they can be collected together to form a color factor C F. A QCD tree-level matrix element can therefore be calculated by looking at the corresponding QED matrix element, substituting the charge and evaluating the color factor. To calculate transition rates, the colour factors have to be averaged over incoming states and summed over outgoing states as for spins. 7 The Weak Force The success of QED and QCD naturally inspires us to try to include a gauge symmetry responsible for the weak force. Charged current processes connect pairs of left-handed fermions, so for the first generation we must have an SU(2) symmetry transformation between the doublets: q L = ( ul d L ) l L = ( νl This is known as weak isospin: the left handed doublets have I W = 1/2 with the upper and lower particle in each having I 3 W = +1/2 and 1/2 respectively. The right handed particles are singlets under this symmetry group and so have I W = 0. The quark kinematic terms of our Lagrangian will be e L ) L q = iq L /D W q L + iq R /D W q R (13) where the weak isospin covariant derivatives D W are defined as in Equation 12 and introduce three new gauge fields, conventionally called W i µ. Equation 13 still maintains its symmetry under U(1) phase rotations such as q L e iy Lθ q L, but this does not have to be the U(1) of electromagnetism and so can couple to neutrinos. We are free to choose the charges for each weak 9

10 multiplet independently, and could even have quarks and leptons transforming under different groups. For aesthetic and practical reasons however, we choose this U(1) group to be that of hypercharge Y, defined in terms of the electromagnetic and weak isospin charges as Q = I W + Y 2 (14) The corresponding gauge field is normally called B µ. The complete covariant derivative in this theory is given by D µ = µ igi W T i W i µ ig 2 Y B µ (15) Note that we have assigned two different, arbitrary coupling constants to the two symmetry groups, and that the three T i generators of SU(2) are the Pauli matrices. We now have a straw man theory of (electro)weak interactions that fails in almost every respect. Not only does it not include EM explicitly, but it fails to predict the phenomenology of the weak force to any degree (massive gauge bosons are still forbidden). Even worse is that a fermionic mass term, e.g. muu = mu L u R + mu R u L explicitly breaks the SU(2) and U(1) symmetries since left- and right-handed fermions couple differently. All fermions have to be massless in this theory! 8 Spontaneous Symmetry Breaking In order to give masses to the fermions, we will have to spontaneously break the SU(2) symmetry by adding a new field with a non-zero vacuum expectation value: the underlying physics is still SU(2) invariant but the vacuum state itself is not. This is the Higgs mechanism. We can arbitrarily add to our theory a new field φ that couples to each fermion with an arbitrary strength G f. This will introduce Yukawa terms into the Lagrangian like G e lφl. 1 We can then specify the following properties of φ: it is a scalar, so the Yukawa terms can produce masses 1 normally written as two terms, one being the hermitian conjugate of the other to ensure that the Lagrangian is real 10

11 it must have weak hypercharge Y = +1 to ensure invariance under U(1) Y, and hence is complex it must be a doublet of weak isospin I W = 1/2 to preserve invariance under SU(2) from Equation 14 φ = ( φ + φ 0 ) The non-yukawa part of the φ Lagrangian will be of the form: and we choose the potential to be L φ = (D µ φ) (D µ φ) V (φ) (16) V (φ) = µ 2 φ φ + λ(φ φ) 2 to give a non-zero expectation value φ φ = µ 2 /2λ = v 2 /2 say. This means that the set of possible ground states of the φ field form a circle in SU(2) space with radius v/ 2, and the vacuum will spontaneously choose one and break SU(2). Global SU(2) U(1) invariance allows us to choose the specific ground state φ 0 = ( 0 v/ 2 Local SU(2) U(1) invariance then allows us to transform all excitations around the ground state to be along the real φ 0 direction, i.e. we can choose φ = 1 ( 2 ) 0 v + H ) (17) 9 Fermion Masses Expanding out the Yukawa terms for the electron gives L e = G ev 2 ee G e 2 ehe The first term gives an electron mass m e = G e v/ 2. The second is an interaction term between the electron and the H field. 11

12 It should be noted that the down quark will acquire mass in a similar fashion, but to give mass to the up quark (and possibly neutrino) requires extra Yukawa terms in the Lagrangian of the form: L d = G u q L φur where φ = ( φ 0 φ + ) 10 Gauge Boson Masses We will now look at the kinematic part of the φ Lagrangian in Equation 16. Using Equation 17, the fact that Y = 1 and I W = 1/2 in Equation 15 and the Pauli matrices as generators of SU(3) it is possible to show that (D µ φ) (D µ φ) = 1 2 µh µ H+ g2 v 2 8 (W 1 +iw 2 ) µ (W 1 iw 2 ) µ + v2 8 (gw 3 g B) The first term is the kinematic term for a new, neutral scalar particle, and the terms we have ignored express the interactions between H and the gauge fields. Defining the linear combination of neutral fields Z = gw 3 g B g 2 + g 2 = cos θ W W 3 sin θ W B where the weak mixing angle θ W is defined by tan θ = g /g, the above expression becomes (D µ φ) (D µ φ) = 1 2 µh µ H + g2 v 2 8 [(W 1 ) 2 + (W 2 ) 2 ] + g2 v 2 8 cos 2 θ W Z 2 The effect of the symmetry breaking is to give the (charged) W 1 and W 2 fields a mass M W = gv/2 and to give a linear combination of W 3 and B an effective mass M Z = gv/2 cos θ W = M W /cosθ w. The orthogonal combination A = sin θ W W 3 + cos θ W B has remained massless. 11 Electroweak Interactions We can express the covariant derivatives in Equation 15 in terms of these new field combinations. For example, the kinematic lepton Lagrangian now 12

13 looks like L l = il /Dl = ie e + iν ν g sin θ W eγ µ ea µ + g cos θ W (sin 2 θ W e R γ µ e R 1 2 cos 2θ W e L γ µ e L ν Lγ µ ν L )Z µ + g 2 (ν L γ µ e W + µ + e L γ µ ν L W µ ) (18) where we have defined the combinations W ± = (W 1 iw 2 ) 2. This gives (in order) kinematic terms for the electron and neutrino a coupling of the electron to the massless gauge field A with strength g sin θ W. Since this couples to both left- and right-handed electrons equally and has no coupling to the neutrino, we can readily identify A as the photon, and deduce that e = g sin θ W a coupling of the left-handed neutrino and both chiralities of the electron to the massive gauge boson Z (all with different coupling strengths) a flavour-changing coupling of left-handed leptons to the massive W gauge boson A similar exercise with the quark fields will give similar results, although the coupling strengths will be different. 12 Standard Model Summary We have developed a theory based on local gauge symmetries. The overall symmetry group is the SU(3) SU(2) U(1) of QCD, weak isospin and hypercharge. The SU(2) U(1) is broken by the addition of a complex scalar field with a non-zero vacuum expectation value. After symmetry breaking, the complete Lagrangian (for one generation) will contain: kinematic terms for the fermions iψ ψ mass terms for the fermions Gev 2 ee etc. interactions between the fermions and the gauge bosons g, A, W ±, Z (see Equations 11 and 18 13

14 mass terms for the W and Z gauge fields interactions between the gauge fields, including self-interactions a new scalar particle, H which interacts with all fermions and massive bosons This describes very well the fundamental particles and their interactions as we observe them in experiment, and a particle consistent with the scalar H or Higgs boson has now been observed. 13 Loops and Renormalisation Going beyond tree-level, Feynman diagrams will contain internal loops. To calculate the corresponding matrix element, a factor of ( 1) n must be introduced for each fermion loop and all internal states summed over. Since an internal loop can have an arbitrary momentum k added to one side as long as the same momentum is subtracted from the other, summing over internal states will result in the appearance of ultraviolet-divergent integrals of the form d 4 k In gauge theories, these integrals diverge logarithmically as k. It should be noted that infrared divergences can also occur but do not cause fundamental problems; for example the number of gluons radiated by a quark approaches infinity as the gluon energy approaches zero, but the number of gluons detectable above any threshold remains finite as does the total energy. Ultraviolet divergences are handled via a process called regularisation. First, the infinite integrals are written as the limiting case of finite integrals e.g. m 2 k 2 dz M 2 z = lim dz M 2 m 2 z Secondly, the divergences are incorporated into redefinitions of the physical parameters. Some divergent terms will appear with opposite signs in different Feynman diagrams and cancel (due to the underlying gauge symmetries via Ward Identities), but some will remain. In electron scattering in QED, second-order corrections to the photon propagator ( vacuum polarisation ) effectively change the charge of the electron giving the QED renormalised coupling strength 14

15 α R (α 0, q 2, m 2, M 2 ). The infrared cutoff m 2 can be made arbitrarily small (as discussed above) with no effect, but the coupling strength remains a function of the photon four-momentum q 2 and the cutoff scale M 2 as well as the fundamental coupling strength α 0. What is actually measured in an experiment is α R, which is now a running coupling, a function of the energy scale at which the measurement takes place. Since α R is finite, in the limit as M 2 the fundamental coupling strength α 0. The mass of the electrons must be similarly redefined in terms of an (infinite) fundamental mass due to loop corrections to the electron propagators. Each order of perturbation theory can, in principle, require a different redefinition of the physical parameters. This would result in a theory with an infinite number of parameters which would be worthless. Fortunately in theories with logarithmic divergencies (such as gauge theories) a single redefinition of parameters will work to all orders in perturbation theory. A theory in which regularisation to all orders can be effected by a finite number of corrections is called renormalisable. In a general theory, a renormalised coupling strength g of dimension [energy] n at an energy scale µ 2 can be expected to depend on the fundamental coupling strength g 0 at some scale Λ 2 according to g g 0 ( µ Λ Theories with n < 0 are renormalisable while those with n > 0 are nonrenormalisable. If n = 0 a theory will be normalisable if the divergencies are logarithmic as is the case with gauge theories. The evolution of g as a function of µ 2 is governed by the β function, defined as β = µ 2 g µ 2 For a logarithmically divergent theory this can be solved to O(g 2 ) as g = g ) n 1 αg log µ2 µ 2 The behaviour of the coupling strength now depends on the coefficient α which can be calculated for a given theory. If α < 0 the coupling strength increases as a function of energy as in the case of QED. For α > 0 the coupling strength increases as energy decreases. This is typical of non-abelian gauge theories and gives rise to the asymptotic freedom and confinement seen in QCD. 15

16 The apparent good luck that the theories describing nature are gauge theories and hence renormalisable may be explicable. If a fundamental theory is assumed at a very large energy scale M 2 with coupling constants naturally of order unity, any non-renormalisable couplings will run quickly down to zero leaving the renormalisable couplings to dominate at our energy scales. For these the interpretation of Feynman diagrams as a perturbation series in terms of the fundamental parameters has to be replaced, as for M 2 q 2 the series clearly diverges. Instead, the Feynman calculus gives the evolution of the running coupling constant about any chosen energy scale µ 2. 16