Weak Interactions – I

Susan Gardner

Department of Physics and Astronomy University of Kentucky Lexington, KY 40506 USA [email protected]

Lecture 1: A survey of basic principles and features with an emphasis on low-energy probes Lecture 2: The electroweak Standard Model — and interpreting it as an effective ﬁeld theory How “Weak” is the Weak Interaction?

We know of four fundamental interactions: electromagnetic, strong, weak, and gravitational. Let’s set gravity aside and focus on the others. Particles of comparable mass can have very different lifetimes.

|gem|2 =⇒ eff ∼ 4 · 10−5 =⇒ |gstr | ∼ 102|gem| str 2 eff eff |geff | Conclude weak interaction is ∼ 106 times weaker than the strong interaction!

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 2 Building a “Standard Model”

A ﬁrst thought: perhaps we can describe all 3 interactions independently? What precepts should we impose? A list circa 1980: Particle Content (i.e., 3 generations of quarks and leptons) Symmetries CPT (and Lorentz) Symmetry Gauge Symmetry (SU(3)×...×...) Unitarity Renormalizability This makes our theory predictive even as E → ∞! It is “UV complete”!

This line of thinking ultimately yields a SU(3)c × SU(2)L × U(1)Y gauge theory we call the Standard Model (SM). It is predictive — and successful — once all its parameters are ﬁxed. In 2015 the existence of known unknowns (e.g., dark matter) is now deﬁnite. Perhaps we can describe these new features within the context of a theory with a SM-like gauge symmetry? Let’s take just “baby steps” beyond the SM!

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 3 on Effective Field Theory

To explain phenomena at some fixed energy scale, we need only include the degrees of freedom operative at that energy scale. E.g., we can predict the outcome of chemical reactions without understanding how the electron gets it mass! A simple application of effective field theory: “Why is the sky blue?” Here we consider low-energy scattering of photons from neutral atoms 2 −1 Eγ ∆E ∼ α me a0 ∼ meα Matom The low-energy interactions of the atom (ψ) are fixed by symmetry: gauge and P and C invariance....

3 † µν Lint = c1a0ψ ψFµν F + ... 3 with the a0 factor ﬁxed by common sense and “power counting” so that µν c1 ∼ O(1).(dim[ψ] = 3/2 and dim[F ] = 2) 3 2 6 4 Thus A ∼ c1a0Eγ =⇒ σ ∼ a0Eγ [~ = c = 1] and we conclude that blue light is scattered more strongly than red light! Thus a theory need not be “UV complete” to be predictive. Through identifying traces of new physics at low energy we hope to identify the nature of E → ∞ physics! We return now to the path that led to the rise of the SM....

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 4 The Discrete Symmetries – C, P, and T

In particle interactions, can we tell... Left from Right? (P) Positive Charge from Negative Charge? (C) Forward in Time from Backward in Time? (T) Matter from Antimatter? (CP)

If we “observed” a box of photons at constant temperature T ∼ me, interacting via electromagnetic forces, the answer would be No.

− e e− However, ... e+ e+

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 5 On the Possibility of Parity Violation

Context: Dirac – the existence of a magnetic monopole can explain the quantization of electric charge! [Dirac, Proc. Roy. Soc. London A 133, 60 (1931)]

∇ · E = 4πρ ; ∇ · B = 0 =⇒ 4πρM Dirac also showed that the circulation of opposite magnetic monopoles in the nucleon could give rise to a nonzero electric dipole moment. [Dirac, Phys. Rev. 74, 817 (1948).] The electric dipole moment d of a nonrelativistic particle with spin S is S deﬁned via H = −d S · E But both quantities violate P and T ! E. M. Purcell and N. F. Ramsey, “On the Possibility of Electric Dipole Moments for Elementary Particles and Nuclei,” Phys. Rev. 78, 807 (1950):

The argument against electric dipoles, in another form, raises the question of parity.... But there is no compelling reason for exclud- ing this possibility....

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 6 Discrete Symmetries — P, T, and C

Parity P: Parity reverses the momentum of a particle without ﬂipping its spin.

s † s s † s † 0 PapP = a−p , PbpP = −b−p =⇒ Pψ(t, x)P = γ ψ(t, −x)

Time-Reversal T : Time-reversal reverses the momentum of a particle and ﬂips its spin. It is also antiunitary; note [x, p] = i~. s † −s s † −s † 1 3 TapT = a−p TbpT = b−p =⇒ T ψ(t, x)T = −γ γ ψ(−t, x)

Charge-Conjugation C: Charge conjugation converts a fermion with a given spin into an antifermion with the same spin.

s † s s † s † 2 ∗ CapC = bp , CbpC = ap =⇒ Cψ(t, x)C = −iγ ψ (t, x)

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 7 The Weak Interactions Violate Parity

There is a “fore-aft” asymmetry in the e− intensity in 60Co~ β-decay.... [Wu, Ambler, Hayward, Hoppes, and Hudson, Phys. Rev. 105, 1413 (1957); note also Garwin, Lederman, and Weinrich, Phys. Rev. 105, 1415 (1957); http://focus.aps.org/story/v22/st19 .] Schematically

ν e e + +

60 60 * ν Co Ni e e (J=5) (J=4)

~ J·~pe Ie(θ) = 1 − Ee P is violated in the weak interactions! BothP andC are violated “maximally”

+ + + + Γ(π → µ νL) 6= Γ(π → µ νR) = 0 ; P violation + + − − Γ(π → µ νL) 6= Γ(π → µ νL) = 0 ; C violation

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 8 The “Two-Component” Neutrino

A Dirac spinor can be formed from two 2-dimensional representations:

ψ ψ = L ψR In the Weyl representation for γµ, µ −m i(∂0 + σ · ∇) ψL (iγ ∂µ − m)ψ = = 0 i(∂0 − σ · ∇) −m ψR

If m=0, ψL and ψR decouple and are of deﬁnite helicity for all p. Thus, e.g., i(∂0 − σ · ∇)ψL(x) =⇒ EψL = −σ · p ψL

σ · pˆ ψL = −ψL ¯ † 0 Note ψL ≡ ψLγ transforms as a right-handed ﬁeld. Experiments =⇒ No “mirror image states”: neither νL nor νR exist. Possible only if the neutrino is of zero mass.

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 9 The Weak Interactions Can Also Violate CP

CP could be a good symmetry even ifP andC were violated. Schematically

ν + e CP e

ν e e

+ + − − Γ(π → µ νL) = Γ(π → µ νR); CP invariance! Weak decays into hadrons, though, can violate CP. There are “short-lived” and “long-lived” K states:

1 0 0 + − KS ∼ √ (K − K ) → π π (CP even) 2

1 0 0 + − 0 KL ∼ √ (K + K ) → π π π (CP odd) 2

However, KL → 2π as well! KS and KL do not have deﬁnite CP! [Christenson, Cronin, Fitch, Turlay, PRL 13, 138 (1964).]

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 10 Matter and Antimatter are Distinguishable

0 ¯ 0 + − 0 ¯ 0 The decay rates for K , K → π π and B , B → J/ψKS are appreciably different.

[I.I. Bigi, arXiv:0703132v2 and references therein.]

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 11 All Observed Interactions Conserve CPT

The CPT Theorem Any Lorentz-invariant, local quantum ﬁeld theory in which the observables are represented by Hermitian operators must respect CPT. [Pauli, 1955; Lüders, 1954] Coda: CPT violation implies Lorentz violation. [Greenberg, PRL 89, 231602 (2002)]

CPT =⇒ the lifetimes, masses, and the absolute values of the magnetic moments of particles and anti-particles are the same! Note, e.g.,

|M 0 − M ¯ | K K0 < 6 × 10−19 @90% CL Mavg

|M − M¯| p p < 7 × 10−10 @90% CL Mp Thus CP ↔ T violation. Tests of CPT and Lorentz invariance are ongoing. e.g., ATRAP Collaboration, arXiv:1301.6310, “... For the ﬁrst time a single trapped p¯ is used to measure the p¯ magnetic moment

µp¯. ... The 4.4 parts per million (ppm) uncertainty is 680 times smaller than previously realized. Comparing to the proton moment

measured using the same method and trap electrodes gives µp¯/µp = −1.000 000 ± 0.000 005 to 5 ppm, for a proton moment

µp = µpS/(~/2), consistent with the prediction of the CPT theorem.” S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 12 Transformations of Lorentz Bilinears under P, T, and C

Notation: ξµ = 1 for µ = 0 and ξµ = −1 for µ 6= 0. 5 0 1 2 3 µν i µ ν γ ≡ iγ γ γ γ ; σ ≡ 2 [γ , γ ]

µ µ µν ψψ¯ iψγ¯ 5ψ ψγ¯ ψ ψγ¯ γ5ψ ψσ¯ ψ ∂µ SPVAT P +1 −1 ξµ −ξµ ξµξν ξµ T +1 −1 ξµ ξµ −ξµξν −ξµ C +1 +1 −1 +1 −1 +1 CPT +1 +1 −1 −1 +1 −1

S is for Scalar P is for Pseudoscalar V is for Vector A is for Axial-Vector T is for Tensor

All scalar fermion bilinears are invariant under CPT.

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 13 Symmetries of a Dirac Theory

A Lagrangian must be a Lorentz scalar to guarantee Lorentz-invariant equations of motion. E.g., applying the Euler-Lagrange eqns to

¯ µ LDirac = ψ(iγ ∂µ − m)ψ

yield Dirac equations for ψ and ψ¯. We can form two currents

jµ(x) = ψ¯(x)γµψ(x); jµ5(x) = ψ¯(x)γµγ5ψ(x)

jµ is always conserved if ψ(x) satisﬁes the Dirac equation:

µ µ µ ∂µj = (∂µψ¯)γ ψ + ψγ¯ ∂µψ = (imψ¯)ψ + ψ¯(−imψ) = 0 ,

µ5 5 whereas ∂µj = 2imψγ¯ ψ — it is conserved only if m = 0. By Noether’s theorem a conserved current follows from an invariance in LDirac: 5 ψ(x) → eiαψ(x); ψ(x) → eiαγ ψ(x) The last is a chiral invariance; it only emerges if m = 0.

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 14 Symmetries of a Dirac Theory

To understand why it is a chiral invariance, we note in the m = 0 limit that

1 − γ5 1 + γ5 jµ = ψγ¯ µ ψ , jµ = ψγ¯ µ ψ . L 2 R 2

The vector currents of left- and right-handed particles are separately conserved. Note in Weyl representation

−1 0 γ5 = 0 1

The factor (1 ± γ5) acts to project out states of deﬁnite handedness.

1 − γ5 1 + γ5 ψ ≡ ψ , ψ ≡ ψ . L 2 R 2

¯ µ ¯ µ so that L = ψLiγ ∂µψL + ψRiγ ∂µψR = LL + LR

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 15 Electromagnetism

We assert that if we couple a Dirac ﬁeld ψ to an electromagnetic ﬁeld Aµ jµ is the electric current density. ψ can describe a free electron. −ip·x µ ψ(x)|p, si = u(p)e =⇒ (γµp − m)u(p) = 0 . By “canonical substitution” pµ → pµ + eAµ

µ 0 0 µ (γµp − m)u = γ Vu ; γ V = −eγµA In O(e) the amplitude for an electron scattering from state i → f is Z Z † 4 fi µ 4 fi ¯ Tfi = −i uf V (x)ui (x) d x = −i jµA d x with jµ = −euf γµui

For e − p scattering, e.g., we have Z 1 T = −i je(x) − jp(x) d 4x = −iM(2π)4δ(4)(p + k − p0 − k 0) ﬁ µ q2 µ

e2 e2 M ≡ − jem (jem µ) = (eu¯ (p0)γ u (p)) − (−eu¯ (k 0)γµu (k)) q2 µ p e p µ p q2 e e A current-current interaction. S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 16 Fermi Theory

− Now consider n → pe ν¯e. Fermi’s crucial insight was to realize that the weak currents could be modelled after electromagnetism: 0 0 µ M = G(u¯p(p )γµun(p))(u¯e(k )γ uν (k)) The observation of e − p capture suggests

GF ¯ ¯ µ LFermi = −√ (ψpγµψn)(ψeγ ψν ) + h.c. 2 An interaction with charged weak currents. A weak neutral current was discovered in 1973. −5 −2 GF is the Fermi constant, though GF ∼ 10 (GeV) . (N.B. not UV complete!!) Suggests the interaction is mediated by massive, spin-one particles. Fermi’s interaction cannot explain the observation of parity violation. Nor can it explain the |∆J| = 1 (“Gamow-Teller”) transitions observed in nuclear β-decay. Some A × A or T × T interaction has to be present. Enter the V − A Law.... [Feynman, Gell-Mann, 1958; Sudarshan and Marshak, 1958] S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 17 Fermi vs. Gamow-Teller Transitions

Nuclear β-decay spin-isospin selection rules are dictated by the form of the nonrelativistic transition operator. In allowed approximation (note πi πf = +1) ... A X τ±(j) = T± “Fermi” =⇒ Jf = Ji , Tf = Ti j=1 Here both parent and daughter are in isotopic analogue states, e.g., 10C →10 B.

A X σ(j) · τ±(j) “Gamow-Teller” =⇒ Ji = Jf , Jf ± 1 (Ji = 0 6→Jf = 0) , j=1 |∆T | = 0, 1

N.B. 0+ → 0+ “superallowed” decays are pure Fermi transitions, whereas neutron β decay is “mixed,” containing both types.

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 18 The V-A Law

A “universal” charged, weak current:

1 GF n λ † † λo l h L = − √ J J + J J with Jλ = j + j 2 2 λ λ λ λ For the leptons... l λ ¯ λ ¯ 0 λ ¯ 0 λ j = ψeγ (1 − γ5)ψνe + ψµ(k )γ (1 − γ5)ψνµ + ψτ (k )γ (1 − γ5)ψντ − + which describes νl → l and l → ν¯l and asserts the leptons do not mix under the weak interactions. Here the “V-A” law is equivalent to a “two-component” neutrino picture. The interactions of the hadrons (quarks) are (and can be) much richer. The strong interaction is strong! The quarks mix under the weak interactions. E.g., K + → µ+ν is observed. Recall K + is (us¯). Let us continue to focus on neutron β-decay. Recall n is ddu and p is uud. Isospin is an approximate symmetry: Mn = 939.565 MeV Mp = 938.272 MeV (Mn − Mp)/Mn 1. − n → pe nu¯ e occurs because isospin is broken =⇒ large τn. S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 19 Polarized Neutron β-decay in a V-A Theory

d 3p 3 3 3 1 p d pe d pν 4 1 P 2 d Γ = 5 ( )δ (pn − pp − pe − pv ) |M| (2π) 2mB 2Ep 2Ee 2Eν 2 spins

GF Vud µ M = √ hp(pp)|J (0)|~n(pn, P)i[u¯e(pe)γµ(1 − γ5)uν (pν )] 2

µ µ f2 µν f3 µ hp(pp)|J (0)|~n(pn, P)i = u¯p(pp)(f1γ − i σ qν + q Mn Mn µ g2 µν g3 µ −g1γ γ5 + i σ γ5qν − γ5q )u~n(pn, P) Mn Mn

Note q = pn − pp and for baryons with polarization P, 1+γ5P/ u~n(pn, P) ≡ ( 2 )un(pn)

f1 (gV ) Fermi or Vector g1 (gA) Gamow-Teller or Axial Vector f2 (gM ) Weak Magnetism g2 (gT ) Induced Tensor or Weak Electricity f3 (gS) Induced Scalar g3 (gP ) Induced Pseudoscalar

Since (Mn − Mp)/Mn 1, a “recoil” expansion is very useful. Also in nuclear decays!

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 20 Correlation Coefﬁcients

3 max 2 d Γ ∝ Ee|pe|(Ee − Ee) × pe · pν pe pν pe × pν [1 + a + P · (A + B + D )]dEedΩedΩν EeEν Ee Eν EeEν A and B are P odd, T even, whereas D is (pseudo)T odd, P even. λ ≡ g1/f1 ≡ gA/gV > 0 and predictions: 1 − λ2 λ(1 − λ) λ(1 + λ) a = A = 2 B = 2 [+O(R)] 1 + 3λ2 1 + 3λ2 1 + 3λ2

2 implying 1 + A − B − a = 0 and aB − A − A = 0 [Mostovoy and Frank, 1976], testing the max V-A structure of the SM to recoil order, O(R) with R ∼ Ee /Mn ∼ 0.0014. Currently

a = −0.103 ± 0.004 A = −0.1184 ± 0.0010 (S = 2.4) B = 0.9807 ± 0.0030

so that the relations are satisﬁed. [λ = 1.2723 ± 0.0023 (S = 2.2)] 2 2 With τn = 880.3 ± 1.1 sec( S = 1.9) and τn ∝ f1 + 3g1 more tests are possible. [Olive et al., Particle Data Group, Chin. Phys. C, 38, 090001 (2014).]

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 21 Symmetries of the Hadronic, Weak Current

The values of the 6 couplings (assuming T invariance) are constrained by symmetry. Conserved-Vector Current (“CVC”) Hypothesis Absence of Second-Class Currents (“SCC”) Partially Conserved Axial Current (“PCAC”) Hypothesis CVC: The charged weak current and isovector electromagnetic current form an isospin triplet. [Feynman and Gell-Mann, 1958]

2 1 Jem ,q = ψ¯ γµψ − ψ¯ γµψ µ 3 u u 3 d d em ,q ¯ µ ¯ µ ψu Jµ = e0ψqγ Iψq + e1ψqγ τ3ψq with ψq = ψd

ψu ψu 0 0 1 τ3 = ; τ3 = − ; e0,1 = (eu ± ed ) 0 0 ψd ψd 2

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 22 Symmetries of the Hadronic, Weak Current

Thus S( 2) F S(q2) em N ¯ S 2 µ F2 q µν 3 µ Jµ = ψ[F1 (q )γ − i σ qν + q ]e0Iψ Mn Mn V 2 V ( 2) ¯ V 2 µ F2 (q ) µν F3 q µ +ψ[F1 (q )γ − i σ qν + q ]e1τ3ψ Mn Mn ψp ψp ψp ψ = and τ+ = ψn ψn 0 The CVC hypothesis implies 2 V 2 2 2 f1(q ) = F1 (q ) and f1(q ) → 1 as q → 0 2 V 2 f2(q ) = F2 (q ) 2 V 2 f3(q ) = F3 (q ) = 0 (current conservation) V V f1(0) = 1 + ∆R ∆R starts in O(α)! [tested to O(0.3%) in 0+ → 0+ decays] f2(0)/f1(0) = (κp − κn)/2 ≈ 1.8529 [tested to O(10%) in A = 8, 12 systems] The Ademollo-Gatto theorem makes the second test more interesting.

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 23 Symmetries of the Hadronic, Weak Current

SCC: “Wrong” G-parity interactions do not appear if isospin is an exact symmetry. [Weinberg, 1958] G ≡ C exp(iπT2) where T2 is a rotation about the 2-axis in isospin space. −ψn exp(iπT2)ψ = −iτ2ψ = ψp

(I) † (I) (I) † (I) GVµ G = +Vµ ; GAµ G = −Aµ “ﬁrst class” (II) † (II) (II) † (II) GVµ G = −Vµ ; GAµ G = +Aµ “second class”

no SCC: g2 = 0 and f3 = 0 (tested to O(10%) in A = 12 system (combined CVC/SCC test)) PCAC: g1/f1 is set by strong-interaction physics: g1(0) fπ Goldberger-Treiman relation = gπNN f1(0) MN Can test some of these relationships through experiments sensitive to recoil-order effects.

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 24 PCAC Tests in Muon Capture

g3 is also predicted by PCAC (HBChPT) and can be studied in µ capture. MuCap redresses the problem with OMC. [Andreev et al., MuCap, PRL 99, 032002 (2007).]

This can also be viewed as a test of lepton-ﬂavor universality, though π (radiative) β decay yields much more severe constraints! Also Kl3, Kl2! S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 25 on Mass

Mass is key to explaining the relative strength of the weak and electromagnetic interactions.

In the SM, the gauge boson masses, as well as those of the elementary fermions, arise through the spontaneously breaking of a local gauge symmetry — the “Higgs mechanism”.

The Higgs mechanism is also key to describing the mixing of the quarks under the weak interactions: it gives rise to the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The Higgs boson has ﬁnally been discovered, and measuring its couplings precisely will tell us whether it is “just” a SM Higgs — or not!

S. Gardner (Univ. of Kentucky) Weak Interactions – I FNP Summer School, UTK, 6/15 26