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129A Lecture Notes
Weak Interactions II
1 Glashow–Weinberg–Salam Theory
We have seen that the weak interaction happens “pretty much” on the isospin lowering or raising operator, except for small admixture of the strange quark. On the side of leptons, they are strictly between e and νe, or μ and νμ. Moverover, the strength of the weak interaction is universal, i.e., it is de- scribed by the same Fermi constant for any processes. This point strongly suggests that the weak interaction is caused by the similar mechanism as the electromagnetism or quark-gluon theory of the strong interaction, where the strength of the force is determined by the overall coupling constant and the charges (or matrices) of given particle type. Based on this motivation, Sheldon Glashow in 1962, and later Steven Weinberg and Abdus Salam, tried to formulate the theory of the weak interaction in terms of the same type of theory, namely gauge theory. Recall the case of the gluon. It is based on three colors, and you demand that you can freely rotate among three colors. This defines three-by-three unitarity rotations, given by the group SU (3) (after removing the overall phase part). There are eight generators for this group, given by Gell-Mann’s lambda matrices (with a factor of 1/2). There are eight gluons, each of which couples to each generator. When the generator acts on a quark, it may change its color. In the same way, we look at the “weak isospin” SU (2). We distinguish it from the “strong isospin” which is between up and down with no admixture of strange, and has nothing to do with leptons. We put together doublets
( u d′
) ,
( νe e
) ,
( νμ μ
)
. (1)
Under the two-by-two unitarity rotation without the overall phase, the SU (2) group, there are three generators given by the three Pauli matrices τ 1 , τ 2 , τ 3 (with a factor of 1/2). Accordingly, we postulate three W -bosons, W 1 , W 2 , W 3. Obviously W stands for “weak.” When they interact with doublets,
they produce a set of amplitudes given by
(¯νμ, μ¯)(g W~ ·
~τ 2
( νμ μ
) = (¯νμ, ¯μ)
g 2
( W 3 W 1 − iW 2 W 1 + iW 2 −W 3
) ( νμ μ
)
. (2)
We introduce the notation W +^ = (W 1 − iW 2 )/
2, W −^ = (W 1 + iW 2 )/
so that it becomes
(¯νμ, μ¯)
g 2
( W 3
√^2 W^ +
2 W −^ −W 3
) ( νμ μ
)
. (3)
The same set is there also for other doublets. Therefore, there are vertices such as μ−^ → νμW −, νe → e−W +, etc. All these vertices come with the universal strength g, analogous to e in the case of electromagnetism. This is very nice. The muon decay, for example, happens because we use the vertex μ−^ → νμW −, where W −^ is virtual. This virtual W −^ has to materialize quickly, such as W −^ → e−^ ν¯e. If you work out numbers, the Fermi constant is then given by GF √ 2
g^2 8 m^2 W
You get two factors of g because you use two vertices, and the amplitude is suppressed by the heaviness of the W -boson. The same can be said about the neutron beta decay, where one of the down-quarks in the neutron emits d → uW −, followed quickly by W −^ → e−^ ¯νe. Because all these are based on the same interaction, no wonder why they come out universal (modulo the mixing of the strange quark). Hurray! We got the theory of the weak force! But it is a little bit too early to celebrate a victory. There are several tricky points which we haven’t seen in other forces we have to deal with. One is that the weak force is of V − A, and hence only the chirality left γ 5 = −1 component couples to it. The way to achieve it is to assume that the right-handed chirality particles are singlets under the weak isospin, so that W -bosons don’t “see” them, in the same way that the photons don’t see neutrinos because of the lack of electric charge. W -bosons couple to “doubletness,” and they do not interact with singlets. Therefore, we have to accept that the particles are organized as
( uL d′ L
) , uR, dR,
( νe eL
) , eR,
( νμ μL
) , μR. (5)
The prefactor is the normalization factor. The photon must be the orthogonal combination, (^) √
g^2 + g′^2 A = g′W 3 + gB. (8)
Instead of always referring to the coupling constants g and g′, it is convenient to introduce the “weak mixing angle,”^2
θW = sin−^1
g′ √ g^2 + g′^2
This allows us to rewrite the Z and photon as simple as
Z = W 3 cos θW − B sin θW , (10) A = W 3 sin θW + B cos θW. (11)
Not only that we mix quarks, we also have to mix gauge bosons. Once we have done this, the interaction of the photon and the Z-boson to the muon (or electron) is fixed uniquely,
1 2
(−gW 3 − g′B) =
(−g(Z cos θW + A sin θW ) − g′(−Z sin θW + A cos θW ))
=
(−g cos θW + g′^ sin θW )Z −
(g sin θW + g′^ cos θW )A.
(12)
The coupling to the photon must be e = −|e|,
|e| =
(g sin θW + g′^ cos θW ) =
gg′ √ g^2 + g′^2
In other words, one combination of two new coupling constants we introduced is already measured, and the unknown parameter is just θW ,
g =
|e| sin θW
, g′^ =
|e| cos θW
(^2) I grew up with people who called it “Weinberg angle.” But Weinberg’s paper actually
does not introduce this angle, but rather uses g and g′^ throughout. It was Glashow who introduced this angle earlier. But θW is a wide-spread notation and we can’t call it Glashow angle anymore. Recently the name “weak mixing angle” is used more widely that solves this problem.
Then the coupling of the left-handed muon (or electron) to the Z-boson is also determined,
1 2
(−g cos θW + g′^ sin θW ) =
|e| sin θW cos θW
( −
)
. (15)
In general, any particle would couple to the combination gI 3 W 3 + g′Y B. Here, I 3 is the third component of the isospin ±^12 for doublets and 0 for singlets. Y is the “weak hypercharge,” which we would like to determine now. By rewriting this combination using the Z-boson and the photon, we find
gI 3 (Z cos θW + A sin θW ) + g′Y (−Z sin θW + A cos θW )
=
|e| sin θW cos θW
(I 3 cos^2 θW − g′Y sin^2 θW )Z + |e|(I 3 + Y )A. (16)
The electric charge of the particle is then given by Q = I 3 + Y. This helps us to determine what we have to take for hypercharges for individual particles. This is the “weak” analogue of Gell-Mann–Nishijima relation. The only possible hypercharge assignement is
( uL d′ L
)+1/ 6 , u+2 R /^3 , d− R^1 /^3 ,
( νe eL
)− 1 / 2 , e− R^1 ,
( νμ μL
)− 1 / 2 , μ− R^1. (17)
The hypercharges are shown as superscripts. Because we have two types of gauge bosons, one coupled to Pauli ma- trices (SU (2)) and the other to numbers (one-by-one hermitian matrices are genetaors, and hence their exponentials are one-by-one-unitarity: U (1)), this theory is called SU (2) × U (1). The hypercharge assignments look very bizzarre, but apart from that, everything works fine. We get the V − A interaction, we understand the univerality of the weak interaction, and we correctly reproduced the photon. On the other hand, we now predict a new force, mediated by the Z-boson, called “neutral-current weak interaction.” We do not know the new parameter, the weak mixing angle yet, and we do not know the masses of the W - and Z-boson yet. But the combination GF /
2 = g^2 / 8 m^2 W = e^2 / 8 m^2 W sin^2 θW is already fixed. Once we know the weak mixing angle, we will know the mass of the W -boson.
was a serious discrepancy between the prediction of SU (2)×U (1) theory and the data, called “high-y anomaly.” But this problem went away by about
- One of the decisive experiments was done at SLAC, the scattering of polarized electrons of deuterium target. You look for the difference in the rate of scattering rate between two polarizations. The scattering is mostly due to the exchange of a virtual photon, which is common to both polar- izations, but a small effect of the virtual Z-boson exchange interferes with the photon-exchange amplitude. The coupling of the left-handed electron is proportional to gZ (I 3 − Q sin^2 θW ∼ − 0 .27), while that of the right-handed coupling to gZ (−Q sin^2 θW ∼ +0.23). Therefore, the relative sign between two amplitudes is the opposite. The difference in the cross section between two polarizations of the electron is proportional to the difference between the left- and right-handed coupling gZ I 3 = e/2 sin θW cos θW. Many mea- surements of sin^2 θW converged, leading to Nobel prize to Glashow, Weinberg and Salam in 1979.
3 Charm
One problem Glashow struggled with was the question of so-called flavor- changing neutral current. We have assigned the up quark and a linear com- bination d′ L ≡ dL cos θC + sL sin θC as a doublet. According to the discussion in the previous section, this combination would have the neutral-current weak interaction,
d¯L^ ′^
( −
sin^2 θW
) d′ LZ (20)
The problem is this. If you write this out in terms of the mass eigenstates dL and sL, you find a new interaction that changes the flavor,
d¯L cos θC
( −
sin^2 θW
) sL sin θC Z (21)
and its complex conjugate. With this interaction, the neutral kaon K^0 (ds¯)
can transform to K 0 (s d¯) at a far too large rate. One of the diagrams is that the d emits a virtual Z and become s, and the Z is absorbed by the ¯s which becomes d¯. Because the Z boson is heavy (91 GeV), this force is very short-ranged, and happens only when d and ¯s basically come to the same point within the size of the kaon. The size of the wave function at the same point can be measured in the process K+^ → μ+νμ, and using the isospin
symmetry, it determines the wave function for K^0. It is characterized by a “kaon decay constant” fK ∼ 160 MeV. (See “Pseudoscalar-Meson Decay Constants” http://pdg.lbl.gov/2002/decaycons_s808.pdf from Particle Data Group for more details.) Then this process would induce
∆m^2 K ∼
g^2 Z m^2 Z
( −
sin^2 θW
) 2 sin^2 θC cos^2 θC f (^) K^2 m^2 K. (22)
This is many orders of magnitudes larger than the observed mass splitting and is completely unacceptable. Because this is a neutral-current weak inter- action that changes flavor, it is called FCNC (flavor-changing neutral current) process. Glashow, Iliopoulos, and Maiani pointed out that this problem of flavor- changing coupling of the Z-boson can be avoided if there exists the charm quark in 1970. The point is that the charm quark belongs to a doublet together with s′^ = −d sin θC + s cos θC , the orthogonal combination from d′. Then the Z-coupling to s′^ also gives a flavor-changing neutral current
d¯L(− sin θC )
( −
sin^2 θW
) sL cos θC Z. (23)
The important point is that this amplitude precisely cancels the contribution from d′^ because of the orthogonality condition. The existence of the charm quark was speculated before them, based on the correspondence between the lepton (two generations had been known since Neddermeyer–Anderson) and the quarks (only up, down, and strange had been known until J/ψ). However, they were the first to point out a necessity of the charm quark. It turns out that the absence of the dangerous Z coupling is not enough to avoid a too-large K^0 - K 0 mixing. There is a second-order weak inter- action called “the box diagram,” where the down and anti-strange quarks in the initial K^0 exchange a virtual W and become up- or charm-quarks,
which again exchange a virtual W to become K 0
. If the up and charm quarks have the same mass, four diagrams with uu¯, u¯c, cu¯, and c¯c cancel exactly again because of the orthogonality condition. However, the charm quark had not been observed and it has be heavier than the up quark if it exists. Once the masses are different, the four diagrams no longer exactly cancel. Therefore, the four diagrams gives a contribution of approximately 1 16 π^2 G
2 F m 2 c cos (^2) θC sin^2 θC f 2 K m 2 K (the prefactor is a typical factor for diagrams with one loop).
Made on 9-Sep-1993 13:15:11 by DREVERMANN with DALI_D1.
ALEPH^ DALI^ Run=15995^ Evt=
Figure 1: Pair production of tau leptons from electron-positron annihilation. One tau decays into an electron and two neutrinos, where the electrons show- ers in the first layer of calorimeter. The other tau decays into three pions and a neutrino where the pions are absorbed in the calorimeter.
fitted to determine mτ = 1776. 96 +0 − 0 ..^1821 +0 − 0 ..^2517 MeV.
5 W and Z Bosons
5.1 Discovery at SppS¯
As we will see later, the W and Z boson masses are predicted in the minimal Standard Model to be mW = 12 gv, mZ = 12 gZ v, where v = (
2 GF )−^1 /^2 =
250 GeV. Once sin^2 θW was measured from the neutral-current experiments, the mass of W and Z were predicted: mW ∼ 80 GeV, mZ ∼ 90 GeV. The masses were so much heavier than any other particles talked about before. The heaviest elementary particle seen by this point was the bottom quark, about 5 GeV. Clearly a new accelerator with an unprecedented energy was needed. They were discovered in CERN proton anti-proton collider called SppS¯. The main technical obstacle behind such a machine was to produce enough anti-protons, and “cool” them to small beams that can be put in accelerators. The cooling technique called stochastic cooling was developed based on the
MEASUREMENT OF THE MASS OF THE 7 LEPTON (^31)
ubject to the requirement 0~ > 2,^ I I of liable V is performed using ], and the maximum likelihood spond to the parameter values
6.96+;:$ MeV, i-
’;:;; ?&
‘4 pb
(12)
Cb)
oted uncertainties are obtained parameters at their maximum ding the parameter values corre- n 1nL of 0.5. inty in rn, is found by setting integrating the likelihood func- onfidence level interval; i.e., for int, rnlow, is defined by
.
J
tnr
Ldm , (^) (13) 0 error point rnhigh is defined by
0.6827 Ldm (^) (14)
ood .function, these error esti- e as those obtained from a de- rocedure embodied in Eqs. (13) e account of any non-Gaussian function in the present analy- ss dependence which is close to a decrease in 1nL of 0.5 yields MeV, so that in this instance ence in the results of the two
s checked by forming th& likeli- sult -21nX = 2.1; in the large d obey a x2 distribution for nine implies that a very good fit has own explicitly in Figs. U(a) and onds to the cross section given rn, = 1776.96 MeV, the mea-
(^1774 1776 ) m, (MeV)
FIG. 18. (a) The c.m. energy dependence of the r+r- cross section resulting from the likelihood fit (curve), com- pared to the data (Poisson errors). (b) An expanded version of (a), in the immediate vicinity of T+T- threshold. (c) The solid cucve shows the dependence of the logarithm of the like- lihood function on rn,, with the efficiency and background parameters fixed at their most likely values; the dashed curve shows the likelihood function from Ref. [3].
IX. SOURCES OF SYSTEMATIC UNCERTAINTY
Five sources of systematic uncertainty are considered: the fitted efficiency parameter e, which, by definition, incorporates the uncertainties in luminosity scale, and also in trigger and detection efficiency (see Sec. VIII); the effective background cross section os; possible bias in the cm. energy scanning procedure; the cm. energy scale; and the spread in cm. energy. The systematic uncertainties associated with the fitted efficiency parameter are obtained by setting E at its +lu
Figure 2: (a) The c.m. energy dependence of the τ +τ −^ cross section. (b) An expanded verision of (a), in the immediate vicinity of τ +τ −^ threshold. (c) The solid curve shows the dependence of the logarithm of the likelihood function on mτ , compared to that from an older work in the dashed line. Taken from “Measurement of the mass of the τ lepton,” BES collaboration, Phys. Rev. D 53, 20–34 (1996), http://cornell.mirror.aps.org/abstract/ PRD/v53/i1/p20_
pμe =
mW 2
(γ + γβ cos θ, sin θ cos φ, sin θ sin φ, γ cos θ + γβ), (29)
pμνe =
mW 2
(γ − γβ cos θ, − sin θ cos φ, − sin θ sin φ, −γ cos θ + γβ). (30)
The point here is that the transverse momentum of the electron and the neutrino,
peT =
√ |p^1 e|^2 + |p^2 e|^2 =
mW 2
sin θ (31)
does not depend on the boost along the beam direction. On the other hand, the pT has a distribution rather than a definite value. It turns out that pT distribution is peaked at its maximum value mW /2. The reason is in a simple phase space factor. The phase space in the W - decay dΩ = d cos θdφ can be rewritten in terms of pT using the relation
above. cos θ =
√ 1 − 4 p^2 T /m^2 W , and hence
dΩ =
4 pT /m^2 W √ 1 − 4 p^2 T /m^2 W
dpT dφ. (32)
The Jacobian is singular at the maximum pT = mW /2 and produces a peak there called “Jacobian peak.” Thanks to this simple kinematics, a large fraction of W events have both electron and neutrino transverse momenta close to the maximum, making the observation easier. The trick then is to build a detector as “hermetic” as possible. A “her- metic” detector covers most of the solid angle around the collision point, so that few particles escape the detector. Basically, you don’t want any “holes.” In practice, you cannot place a detector along the beam axis because they get burnt too quickly. You don’t want to disturb the beam either. But you don’t care so much about having particles escaping along the beam direction, because they carry little transverse momentum. As long as you are looking for signals based on the transverse momentum, you lose little transverse mo- mentum in your event due to the hole along the beam direction. Then you sum up all the transverse momentum you have observed as vectors in (x,y) plane, and ask how much you don’t see, i.e., the “missing transverse mo- mentum.” You regard this quantity to be the transverse momentum of the neutrino. Therefore, you look for an energetic electron with large transverse momentum, and also for a large missing transverse momentum. UA1 collab- oration lead by Carlo Rubbia showed that looking for each of them result in
the same set of events, namely a set of events with both a high-pT electron and a large missing pT. They had five events all together when they reported the discovery of the W -boson. And the largest pT in this set of events agreed with roughly 40 GeV, consistent with the expectation. For the discoveries of W and Z boson, van der Meer and Rubbia shared the Nobel prize in 1984. In more recent high-statistics samples gathered at Tevatron, it is more customary to look at the “transverse mass,” defined by
mT =
√ (peT + pνT )^2 − (~peT + ~pνT )^2. (33)
This is analogous to the definition of the usual mass m =
E^2 − ~p^2 , except that only transverse quantities are used and hence is boost invariant along the beam direction. It does not necessarily assume that the p~T e^ + p~T ν^ is strictly zero either, because it is there due to the Fermi motion of partons. Using the idealized limit again with no Fermi motion, this quantity is mT = mW sin θ, and the distribution again has the Jacobian peak. It is smeared beyond mW due to the resolution effect. The current world average is mW = 80. 423 ± 0 .039 GeV. Just a word on nomenclature. Because the electron kinematics is mea- sured mostly using the calorimeter, it is customary to use ET = E sin θ for electrons. On the other hand, the muon kinematics is measured using the tracking detector, and we use pT. They are of course the same for nearly massless particles such as electrons and muons, but this is the notation used in the literature. The UA1 experiment has also discovered the Z boson using its decay Z → e+e−^ and Z → μ+μ−. This case, you can use the full four-momentum information because you detect both decay products with no missing mo- menta. A much improved data set from Tevatron is shown below. The Fig. 5 shows lego plot of the W events. This type of plot is called lego-plot, showing the energy deposit in the calorimeter after opening the cylinder. The long direction is the beam direction, while the short one the azimuth around the beam. The beam direction is shown in terms of the pseudo-rapidity
η =
log
1 + cos θ 1 − cos θ
= − log tan
θ 2
which is useful because it only shifts under the boost. We can see this as follows. For massless particles, the pseudo-rapidity coincides with the
0
50
100
150
200
250
70 75 80 85 90 95 100 105 110 Mee (GeV/c^2 )
Events / 1 GeV/c
2
(a) Z → e
e
2
/ dof = 1.
Figure 4: Invariant mass distribution. The points are the data, and the solid line is the Monte Carlo simulation (normalized to the data) with best fit. Taken from “Measurement of the W Boson Mass with the Collider Detector at Fermilab,” Phys. Rev. D64, 052001 (2001), http://link.aps.org/ abstract/PRD/V64/E052001/.
rapidity
y =
log
E + pz E − pz
Under boost, E → Eγ + pz γβ, pz → pz γ + Eγβ. Therefore,
y →
log
E(γ + γβ) + pz (γβ + γ) E(γ − γβ) + pz (γβ − γ)
= y +
log
1 + β 1 − β
Therefore the seperation in pseudo-rapidities among energy deposits in a single event is boost invariant. In this event, the fact that there is only one dominant energy deposit means there is a large missing transverse energy. In the similar lego plot for the Z-event, however, two energy desposits appear at azimuths different by 180◦, and hence back-to-back in the transverse plane. There is no apparent missing transverse momentum.
5.2 LEP
When LEP (Large Electron Positron collider) at CERN, Geneva, started to produce millions of Z-boson, it became possible to measure the Z coupling to different particle species directly.
e
ET ≅ 41 GeV
Figure 5: Event display of pp¯ → W followed by W → eνe at CDF, Tevatron. The parton-level process is u d¯ → W +^ etc.
Emax = 60.2 GeV
e
e
ET ≅ 44 and 36 GeV
Figure 6: Event display of pp¯ → Z followed by Z → e+e−^ at CDF, Tevatron. The parton-level process is u¯u → Z etc.
Figure 7: Magnetic field evolution measured in the tunnel by NMR8. The field increase during this period shows variations of the slope and steps of various sizes. Taken from Nucl. Instrum. Meth. A417, 9-15 (1998).
dσ d cos θ
(e− Re+ L → μ− L μ+ R) ∝ |geR|^2 |gμL|^2 (1 − cos θ)^2 , (39) dσ d cos θ
(e− Re+ L → μ− Rμ+ L ) ∝ |geR|^2 |gμR|^2 (1 + cos θ)^2. (40)
Here, gLe = gμL = (I 3 e − Qe sin^2 θW ) = −^12 + sin^2 θW , geR = gRμ = −Qe sin^2 θW = sin^2 θW are the couplings of the particle species to the Z-boson (with the overall size gZ stripped because it is common to all processes). If you look at the total “forward” events, namely cos θ > 1, the total will be given by
∫ (^1)
0
d cos θ
dσ d cos θ
(e−e+^ → μ−μ+)
∝
(|gLe|^2 |gLμ|^2 + |gRe|^2 |gμR|^2 ) +
(|geL|^2 |gμR|^2 + |gRe|^2 |gLμ|^2 ), (41)
and the backward events ∫ (^) −
− 1
d cos θ
dσ d cos θ
(e−e+^ → μ−μ+)
(|gLe|^2 |gLμ|^2 + |gRe|^2 |gμR|^2 ) +
(|geL|^2 |gμR|^2 + |gRe|^2 |gLμ|^2 ). (42)
Figure 8: LEFT: The LEP ring surrounded by the French and Swiss railroads with the locations of the NMR probes and the four experiments. Two probes are installed in a reference dipole magnet (a), which is connected in series with the LEP dipoles. NMR4 (b) and NMR8 (c) are mounted directly in LEP dipoles in the tunnel. RIGHT: Train leakage currents, vacuum chamber currents and the associated magnetic field perturbation on Nov. 13th, 1995. The observed peaks are coincident with the departure of the 16:50 Geneva– Paris TGV (SNCF). Taken from Nucl. Instrum. Meth. A417, 9-15 (1998).
(^087 88 89 90 91 92 93 94 95 )
5
10
15
20
25
30
35
40
σ^ (nb)
= E cm (GeV)
2 ν's 3 ν's 4 ν's (^) L
ALEPH DELPHI OPAL
√^ s
Figure 9: The lineshape of the Z-boson. Data vs calculations for various numbers of neutrinos.
