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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.
π + → µ+ νµ
π 0 → 2γ
[99.98% of all π + decays] ;
τπ+ ∼ 2.6 · 10−8 s
[98.8% of all π 0 decays] ;
τπ0 ∼ 8.4 · 10−17 s.
Γ ∝ τ −1 =⇒
em 2
|
|geff
em
weak
∼ 108 =⇒ |geff
| ∼ 104 |geff
|
weak 2
|geff |
whereas
ρ0 → π + π −
0
+ −
ρ →µ µ
=⇒
[∼ 100% of all ρ0 decays]
[∼ 4.6 · 10−5 of all ρ0 decays]
em 2
|geff
|
str
em
∼ 4 · 10−5 =⇒ |geff
| ∼ 102 |geff
|
str 2
|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
Eγ ∆E ∼ α2 me a0−1 ∼ me α Matom
The low-energy interactions of the atom (ψ) are fixed by symmetry: gauge
and P and C invariance....
Lint = c1 a03 ψ † ψFµν F µν + . . .
with the a03 factor fixed by common sense and “power counting” so that
c1 ∼ O(1). (dim[ψ] = 3/2 and dim[F µν ] = 2)
Thus A ∼ c1 a03 Eγ2 =⇒ σ ∼ a06 Eγ4 [~ = 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+
S. Gardner (Univ. of Kentucky)
e−
However, ...
e+
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
defined via H = −d SS · 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 excluding 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
Paps P † = a−p
s
Pbps P † = −b−p
,
=⇒ Pψ(t, x)P † = γ 0 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
Taps T † = a−p
−s
Tbps T † = b−p
=⇒ T ψ(t, x)T † = −γ 1 γ 3 ψ(−t, x)
Charge-Conjugation C:
Charge conjugation converts a fermion with a given spin into an
antifermion with the same spin.
Caps C † = bps
S. Gardner (Univ. of Kentucky)
,
Cbps C † = aps
=⇒ Cψ(t, x)C † = −iγ 2 ψ ∗ (t, x)
Weak Interactions – I
FNP Summer School, UTK, 6/15
7
The Weak Interactions Violate Parity
~ β-decay....
There is a “fore-aft” asymmetry in the e− intensity in 60 Co
[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
νe
+
Ni *
60
60
(J=5)
(J=4)
Co
e
+
e
Ie (θ) = 1 −
νe
~
J·~
pe
Ee
P is violated in the weak interactions!
Both P and C are violated “maximally”
Γ(π + → µ+ νL ) 6= Γ(π + → µ+ νR ) = 0
+
+
−
−
Γ(π → µ νL ) 6= Γ(π → µ ν L ) = 0
S. Gardner (Univ. of Kentucky)
Weak Interactions – I
; P violation
; C violation
FNP Summer School, UTK, 6/15
8
The “Two-Component” Neutrino
A Dirac spinor can be formed from two 2-dimensional representations:
ψL
ψ=
ψR
In the Weyl representation for γ µ ,
−m
(iγ µ ∂µ − m)ψ =
i(∂0 − σ · ∇)
i(∂0 + σ · ∇)
−m
ψL
ψR
=0
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 · p̂ ψL = −ψL
ψL† γ 0
Note ψ̄L ≡
transforms as a right-handed field.
Experiments =⇒ No “mirror image states”: neither ν L 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 if P and C were violated.
Schematically
νe
CP
e
Γ(π + → µ+ νL ) = Γ(π − → µ− ν R )
e+
νe
; CP invariance!
Weak decays into hadrons, though, can violate CP.
There are “short-lived” and “long-lived” K states:
1
0
KS ∼ √ (K 0 − K ) → π + π − (CP even)
2
1
0
KL ∼ √ (K 0 + K ) → π + π − π 0 (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
The decay rates for K 0 , K̄ 0 → π + π − and B 0 , B̄ 0 → 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.,
|MK 0 − MK̄0 |
< 6 × 10−19 @90% CL
Mavg
|Mp − Mp̄ |
< 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 p̄ is used to measure the p̄ magnetic moment
µp̄ . ... 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 µp̄ /µp = −1.000 000 ± 0.000 005 to 5 ppm, for a proton moment
µp = µp S/(~/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: ξ µ = 1 for µ = 0 and ξ µ = −1 for µ 6= 0.
γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 ; σ µν ≡ 2i [γ µ , γ ν ]
P
T
C
CPT
ψ̄ψ i ψ̄γ5 ψ ψ̄γ µ ψ ψ̄γ µ γ5 ψ ψ̄σ µν ψ ∂µ
S
P
V
A
T
+1
−1
ξµ
−ξ µ
ξµξν
ξµ
+1
−1
ξµ
ξµ
−ξ µ ξ ν −ξ µ
+1
+1
−1
+1
−1
+1
+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
LDirac = ψ̄(iγ µ ∂µ − m)ψ
yield Dirac equations for ψ and ψ̄.
We can form two currents
j µ (x) = ψ̄(x)γ µ ψ(x)
;
j µ5 (x) = ψ̄(x)γ µ γ 5 ψ(x)
j µ is always conserved if ψ(x) satisfies the Dirac equation:
∂µ j µ = (∂µ ψ̄)γ µ ψ + ψ̄γ µ ∂µ ψ = (imψ̄)ψ + ψ̄(−imψ) = 0 ,
whereas ∂µ j µ5 = 2imψ̄γ 5 ψ — it is conserved only if m = 0.
By Noether’s theorem a conserved current follows from an invariance in
LDirac :
5
ψ(x) → eiα ψ(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
ψ , jRµ = ψ̄γ µ
ψ.
jLµ = ψ̄γ µ
2
2
The vector currents of left- and right-handed particles are separately
conserved.
Note in Weyl representation
−1 0
5
γ =
0 1
The factor (1 ± γ 5 ) acts to project out states of definite handedness.
1 − γ5
1 + γ5
ψL ≡
ψ , ψR ≡
ψ.
2
2
so that L = ψ̄L iγ µ ∂µ ψL + ψ̄R iγ µ ∂µ ψR = 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
j µ is the electric current density. ψ can describe a free electron.
ψ(x)|p, si = u(p)e−ip·x =⇒ (γµ pµ − m)u(p) = 0 .
By “canonical substitution” pµ → pµ + eAµ
(γµ pµ − m)u = γ 0 Vu
;
γ 0 V = −eγµ Aµ
In O(e) the amplitude for an electron scattering from state i → f is
Z
Z
†
4
Tfi = −i uf V (x)ui (x) d x = −i jµfi Aµ d 4 x with jµfi = −eūf γµ ui
For e − p scattering, e.g., we have
Z
1
Tfi = −i jµe (x) − 2 jµp (x) d 4 x = −iM(2π)4 δ (4) (p + k − p0 − k 0 )
q
M≡−
2
e2 em em µ
e
0
j
(j
)
=
(e
ū
(p
)γ
u
(p))
− 2 (−eūe (k 0 )γ µ ue (k ))
p
µ p
e
q2 µ p
q
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− ν̄e .
Fermi’s crucial insight was to realize that the weak currents could be
modelled after electromagnetism:
M = G(ūp (p0 )γµ un (p))(ūe (k 0 )γ µ uν (k ))
The observation of e − p capture suggests
GF LFermi = − √ (ψ̄p γµ ψn )(ψ̄e γ µ ψν ) + h.c.
2
An interaction with charged weak currents.
A weak neutral current was discovered in 1973.
GF is the Fermi constant, though GF ∼ 10−5 (GeV)−2 . (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.,
10
C →10 B.
A
X
σ(j) · τ± (j) “Gamow-Teller” =⇒ Ji = Jf , Jf ± 1 (Ji = 0 →J
6 f = 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:
o
1 GF n λ †
L=− √
J Jλ + Jλ† J λ
2 2
For the leptons...
with Jλ = j l λ + j h λ
j l λ = ψ̄e γ λ (1 − γ5 )ψνe + ψ̄µ (k 0 )γ λ (1 − γ5 )ψνµ + ψ̄τ (k 0 )γ λ (1 − γ5 )ψντ
which describes νl → l − and l + → ν̄l 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 (u s̄).
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− n̄ue 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 3Γ =
d 3 pp d 3 pe d 3 pν
1
4
(2π)5 2mB ( 2Ep 2Ee 2Eν )δ (pn
M=
− pp − pe − pv ) 12
P
spins
|M|2
GF Vud
√ hp(pp )|J µ (0)|~n(pn , P)i[ūe (pe )γµ (1 − γ5 )uν (pν )]
2
f2 µν
f3 µ
σ qν +
q
Mn
Mn
g2 µν
g3
−g1 γ µ γ5 + i
σ γ5 qν −
γ5 q µ )u~n (pn , P)
Mn
Mn
hp(pp )|J µ (0)|~n(pn , P)i = ūp (pp )(f1 γ µ − i
Note q = pn − pp and for baryons with polarization P,
/
5P
u~n (pn , P) ≡ ( 1+γ
2 )un (pn )
f1 (gV ) Fermi or Vector
f2 (gM ) Weak Magnetism
f3 (gS )
Induced Scalar
g1 (gA )
Gamow-Teller or Axial Vector
g2 (gT ) Induced Tensor or Weak Electricity
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
d 3 Γ ∝ Ee |p e |(Eemax − Ee )2 ×
p ·p
p
p
p × pν
[1 + a e ν + P · (A e + B ν + D e
)]dEe dΩe dΩν
Ee Eν
Ee
Eν
Ee Eν
A and B are P odd, T even, whereas D is (pseudo)T odd, P even.
λ ≡ g1 /f1 ≡ gA /gV > 0 and predictions:
a=
1 − λ2
1 + 3λ2
A=2
λ(1 − λ)
1 + 3λ2
B=2
λ(1 + λ)
1 + 3λ2
[+O(R)]
implying 1 + A − B − a = 0 and aB − A − A2 = 0 [Mostovoy and Frank, 1976], testing the
V-A structure of the SM to recoil order, O(R) with R ∼ Eemax /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)]
With τn = 880.3 ± 1.1 sec (S = 1.9) and τn ∝ f12 + 3g12 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]
Jµem ,q =
2
1
ψ̄u γ µ ψu − ψ̄d γ µ ψd
3
3
Jµem ,q = e0 ψ̄q γ µ Iψq + e1 ψ̄q γ µ τ3 ψq
τ3
ψu
0
=
ψu
0
S. Gardner (Univ. of Kentucky)
;
τ3
0
ψd
with ψq =
=−
Weak Interactions – I
0
ψd
ψu
ψd
;
e0,1 =
1
(eu ± ed )
2
FNP Summer School, UTK, 6/15
22
Symmetries of the Hadronic, Weak Current
Thus
F S (q 2 ) µ
F2S (q 2 ) µν
σ qν + 3
q ]e0 Iψ
Mn
Mn
F V (q 2 ) µ
F V (q 2 ) µν
+ψ̄[F1V (q 2 )γ µ − i 2
σ qν + 3
q ]e1 τ3 ψ
Mn
Mn
ψp
ψp
ψp
ψ=
and τ+
=
ψn
ψn
0
Jµem N = ψ̄[F1S (q 2 )γ µ − i
The CVC hypothesis implies
f1 (q 2 ) = F1V (q 2 ) and f1 (q 2 ) → 1
2
f2 (q ) =
f3 (q 2 ) =
F2V (q 2 )
F3V (q 2 )
=0
as q 2 → 0
(current conservation)
f1 (0) = 1 + ∆VR
∆VR 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 = −iτ2 ψ =
ψp
GVµ(I) G† = +Vµ(I)
GVµ(II) G† = −Vµ(II)
†
(I)
GA(I)
µ G = −Aµ
;
;
†
(II)
GA(II)
µ G = +Aµ
“first class”
“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:
(0)
= gπNN MfπN
Goldberger-Treiman relation gf11(0)
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