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_
-Lg suge -
-
'4I W.~lJ WIa ~v -
I B
4"
Bl'v
(15.4)
wbw, J1
(15.5)
IHI
where
• W JW
=
~
VJi
W"v
-
iJ·v W· iJ. - 9 1alK'
II
as defined in Eq. (14.20) for general Yang-Mills fields. We have put a dagger in one of the factors of W~v in Eq. (15.4). This is because the index a, b, c take the values +, -,3, and fabc are not real in this basis. For the U(I) part of the theory, the definition of B~v is similar to that of the field tensor for electromagnetism: (15.6) As was observed for a general Yang-Mills theory and for QED, the pure gauge Lagrangian cannot contain allY mass term for the gauge bosons since such a term would not be gauge inva.riant, However, experiments show that the gauge hosons nwdialing weak interactions are massive.
Chapter 15. Standard electroweak theory
330
15.2
Spontaneous symmetry breaking
15.2.1
Introducing the Higgs boson multiplet
The way to solve this problem was indicat.ed in §13.6, where we showed how spontaneous symmetry breaking in a gauge theory leads to the generation of mass for gauge bosons. To implement the same
idea in the present cont.ext, we need some sca.lar fields in the theory. Let us introduce a .calar multiplet which is a doublet under the SU(2) part of the gauge group, and write this as
(15.7) On the right, we have ::;l1ow11 the gauge transformation properties of
this multiplet. The '2' indicates that it i. a doublet of SU(2), and we have normalized the U(I) charge such t.hat its value is ~ for the multiplet ¢. The Lagrangian now contains involving ¢ ,1."\ well. These are (15.8) Here Dp.. as before, is the gauge covariant derivative. Since the gauge group l.lOW contains two factors, we should have the gauge bosons of both factors ill the definition of D JJ, and these two sets can corne with two different coupling constants. In general, when a multiplet transforms like an n-dimensional representation of SU(2) and has a U(I) quantum number Y, we should write
DJ.l =
a
JJ
. T(n)w . 'YB Ii + 1.9 a I.l + 19 a
I
(15.9)
where T!") deuote the generators of SU(2) in the n-dimensional rej>resentation, and Y is the identity matrix times the hypercharge. For the doublet ¢, the SU(2) generators are Ta /2, and we can write
(::)
.
(15.10)
15.2.
SpolJtaneous sym1IJetry breaking
331
Let us now suppose, as in various cxamples 011 spoutallcous symmetry breaking in Ch. 13, that 1,2 < 0 in Eq. (15.8). In this case, the gauge sYlllIllctry will bc spontaneollsly brok~l\. The minimuJII for the scalar potential will be ohtained for ~calar ricld COil figurations given
by v
rt>"
(15.11)
= /2 X ,
where X is a douhlet sati,fying XIX = I, and
,,=/_,,2/>,.
(15.12)
vVithout a.ny loss of generality, we will consider our system to be in the vacuum sta.te
(15.13)
\.e., X
= (~).
Any other vacu II rn ,atisfvinJl E'1' (15.11) can be
reachcd from this by a Jllobll.1 8U(2) x vacuum state, the generators T 1 and symmctry, since Tx 1>" # 0, T y 1>c1 # O. and the U(I) generator Y abo do not
U(I) t.ran,furlllatiou. In this T'). are no more part of the Tlw dial(ollal gencrat.ors T3 annihilate the VaCUUm state.
However I notice that
(15.14) Thus there is one diagonal gencrator which annihilates tile vacuum. This is a linear combinat,ioll of T3 and Y 1 givell b.y
(15.15) The original gauge ,yrnmetry is therefore hroken down to a U(I) sYIIlTlletry gencrated hy Q. Thi, is not the originaJ U(I) part of the symlllctry group. As we have shown, its generator j::; actually a combination of an 8U(2) generator and the original U(I) generator. To make this distinction, we will write the origillal sYlllmetry froUl Oil
flOW
;c, 8U(2) x U(I)y. wherea., thc rellillall\' s.Vlllllletry will be called
lI(l)Q. TIll' lat.ter is in fact the electrOlU
Chapter 15. Standard electroweak theory
332
as we will see fro!)l the couplings of its gauge boson with fer!)lions. For now, notice that the superscripts on the cO!)lponents of '" which appeared in Eq. (15.7) are indeed the electric charges of the corresponding fields.
15.2.2
Gauge boson masses
Since 4>0 cannot be described by creation anel annihilation operators, we can use the technique employed in Ch. 13 and write (15.16)
where 4>+, Hand (are all quantum fields with vanishing expectation values in the vacuum state. Using this decomposition. we can write the Lagrangian of Eq. (15.8) in tenns of the quantum fields. Using the expression for D,,4> from Eq. (15.10) and ",hstituting Eq. (15.16) in it, we find various kinds of . Oue of thcse pick up only the VEV part of the. scalar field 4>. The.e quadratic in the gauge fields are
where
2'2 =
~ (gW,~+9IB"
IIxI1 2 =
XIX for any column matrix ,t. Explicitly. this gives
2
_
2'2 -
J2gW;
J2gW,:
-gW~ + g' fl"
) (
~ )
2
72
1 '.! ,:\ 2 41 g 2 v 2 w,.+W" _ + 8" (-gU ,. + 9 fl,,) . I
(15.17)
(15.18)
For any charged spiu-l field V;, of mass M. the. milSs term in the Lagrangian is A,f2~rVJ'. Thus, the IlHl.5S of the charged W' -boson is 1
MW=2 gv .
(15.19)
-gW; + g' Bj.J of neutral gauge bosuns appearing in Eq. (15.18), we first define the combin"-tions
As for the combination
J Z" = cos Ow Wj.J - sin8w DIJ
A"
= sin Ow
w;: + ern; Ow B
1
I, .
(15.20)
15.2. Spontaneous symmetry breaking
333
where Ow is called the Weinberg angle, defi ned by g' tanOw=-.
(15.21)
9
Then the kinetic for Z~ and A~ are properly normalized because this is an orthogonal transformation, and the last term of Eq. (15.18) can be written as kv2(g2 + gfl)Z~Z~. Since the mass term for a real spin-1 field v~ is ~M2V~ V~, we conclude that the mass of the Z particle is
(2
1 M z=-g+g 2
'2) y, v=
Mw . cos Ow
(15.22)
We see that A~ remains massless even after the symmetry breaking process. As we mentioned, only a part of the gauge symmetry is broken. There is "remaining V(l) symmetry. A~ is the gauge boson corresponding to this symmetry. In §15.3, we will see that this gauge boson is in fact the photon, and so the remnant symmetry is the V(l) symmetry of QED.
15.2.3
Scalar modes
In §13.6, we discussed that a gauge boson can become massive in a spontaneously broken theory by eating up a scalar degree of freedom. This scalar disappears altogether from the theory in the unitary gauge. In other gauges, it appears with a propagator containing an unphysical pole and therefore represents an unphysical mode. In the electroweak theory so far, we have seen that three gauge bosans, the W+ I W- and the Z I become masSive. There must he three unphysical modes corresponding to them. One way to identify these modes is to look at the 4>-Lagrangian in Eq. (15.8) and substitute the expression for in of the quantum fields which appears in Eq. (15.16). Among other things, we will find like
(15.23) As argued in §13.6, this shows that the W+ can be annihilated at a point with the creation of a ¢+, without any ot.her particle interacting with them. This shows t.hat·+ is indeed the unphysical mode eaten up by the W in the process of symmetry breakin!?;. Similarly, W-
334
Chapter 15. SUlIulard electroweak theory
eats up -, and Z eats up the field (. We can see this clearly if we go to the unitary gauge by writing t.he doublet cb in a manner similar to what wa., done in Eq. (13.66). In sneh a gange, only the observable degrees of freedom appear in the Lagrangian. and the particle interpretation is straight forward. o Exercise 15.2 ShOUl that Eq. (15.8) tuhich inuolve n
~:S'i.nglt'.
(15.2~l) ("()'fllnllls
uLl te.rnt3 in Eq.
rleri\Jutille.
o Exercise 15.3 * Construct thr, unitury gn.uge. explir.itly blJ writing th~ doublet 1> in. tc.rms oj quantum, tichb l:\O t1lut after a. ga.uge tTansforTllation the un:physir,ul de.grt',E'.s oj jTer-dam disappe.ur jrom the
Lagrangian.
For calculational purposes, however, it is more cOllvenient to usc a
gauge fixing term snch tilllt the mixed quadratic of Eq. (15.23) disappear from the Lagrangian. As we discussed in §13.6, this can be achieved if we choose the gauge fixing term to be (15.24) The of Eq. (15.23) now pair witli the cm'" from this equation to become tota.l divergence . awl we are left with the
gauge boson propagators in the form W)
( iD,," (10) ~
( Z)
iD,,~ r
=
-I.
10 2 -
Ma,
[
+ iE
-i (10) = 102 _ Ml+
''" -
[ iE
9
9
,w -
(1 - 010,,10"] 1: 2 - t:Ma,
(15.25)
Ok",:,,] '2
(15.26)
(I k" - t: M
The unphYRical ~calar propagators a.re similar t.o that ohtained in Eq.
(13.76): i~($'l(k) = i~(()(k) =
I
102 . -t:,\["\I' . I
10 2 -
t:AI'j, .
(15.27) (15.28)
This choice of gallg" is called t.he R, gaug". Th" 011 bscript. t: in the name refers to the gauge parameter: and tht, [t'tter H indicates that the power-counting rC'lIonnali:tauility of the theory i::; obvious ill this gauge since all propa~at.ors seale like 11k'}. for largE" vallles of k.
15.3. Fermions in the theory
335
There is also a physical scalar mode called the Higgs boson, which corresponds to the qUi\ntum field H in Eq. (15.16). It is an uncharged scalar boson with mass given by
M~ = 2.\v 2 .
(15.29)
Existence of this Higgs boson can be counted ",nong the definitive predictions of the standard model.
15.3
Fermions in the theory
When the electroweak theory was first formulated. it. appeared only as a theory of leptons. Although subsequently the quarks were included in the framework and it is now known that. the theory suffers from some inconsistencies without the quarks. it is simplest to discuss first the leptonic sector of the theory without bringing. the quarks in. Moreover, we will start with only the electron and the electron-neutrino, leaving the other leptons and quarks for §15.3.4.
15.3.1
Gauge interactions
The electron and the electron-neutrino form what is called the first generation of leptons,. Parity is known to be violated in weak interactions, so it is expected that the different chiral components will have diffcrcnt interactions. In the standard c1ectroweak model, these fermions are supposed to transform like the following gauge multiplets: WeI. -
(~~,) e/l
1 (2. -2)'
(1,-1).
(15.30)
It should be noted that we have written dowli both the left and right chiral projc'Ctions of the electron. but olily the left chiml projection for the neutrino. At the time of the formulation of the theory and for quite a while afterwards, all laboratory experiments showed that neut.rinos always appeared to be left halld"d. and t.hat is why t.he right chiral neutrino .state was assumed not to exist ill the standard model. As we will see, a consequence of thi:-> assumption is that the neutrinos turn out to be ma.."islcS-';, III !lI0[P' recent timeoS there
Chapter 15. Standard electroweak theory
336
has been some indication that neutrinos may have mass. If these observations are confirmed, some modification of the standard model would be necessary. It is not clear what the correct modification would be. Howeverl neutrino masses, even if non-zero, must be small. For most processes which do not depend crucially on a non-zero neutrino mass, one can still obtain a good estimate by taking the neutrinos to be massless. For this reason, we present here the original version of the standard model, without any possible modifications which might be prompted by non-zero neutrino masses. We can write down the gauge interactions for tne leptons. The gauge covariant kineti<: for the leptons are
it' = W•. Li'Y~ D~oJ!'L
+ eRi-y~ D"c/i.
(15.31)
D,.
The general form for was given in Eq. (15.9). Since eR is an SU(2) singlet, the SU(2) generators Tn = 0 for it. Using Q = T 3 + Y, we see that both eL and eR have electric charge -1, while V,L is uncharged. This justifies the hypcrcharge assignments for these fields. o Exercise 15.4 ShOUl that the. ITec. Lugrun.yinn oj a Dirac jield can. be written in tcrm~ of the c.hirol projt'..etioTt~ ILS .!L' = 'hh"a",pt
+ ,pRh"a~,pR - m('h,pR + ,pR,pd· (15.32)
The derivative tcrllls in Eq. (15.31) give the usual kinetic for the elect rOll and the neutrino fields. The other in Eq. (15.31) are the interaction of the leptons with the gauge bosons:
~nl
-\IIcL'Y /1 (g7 (1 2
W,~ - ~dBJ.i)
9 (" w+ -J2 veL'YeL I!
1JJt'f, +eR'YJ1.g'BJ1.eR
+ -el,'Y " I/r:f"'l \1'-) t
1_ J1. 3 I -:iVd,'! v,dgW~ - 9 B,,)
1 3 +:ien~Cf,(9W~
+ 9' B") + '1I)"g' B"ell.
(15.33)
The interactions involving W± are cal1~d r;harged current interactions because these gauge hosons arc elect.rically charged. These lead to the following Feynmall rules: W"
-34 JLL ';2'Y ,
(15.34)
15.3. Fermions in the theory
337
!(1-
'Ys), the projection operator for left chirality defined where L = in §4.7. The incoming fermion line is V e and the outgoing one electron if the gauge boson going out is W+, and the converse if the gauge boson going out is W-. Let us now look at the interactions of neutral gauge bosons in Eq. (15.33). These are called neutml current intemctions, and we should rewrite them using the neutral boson eigenstates Z" and the photon. Looking back at the definition of the Z boson in Eq. (15.20), we see that neutrinos have neutral current interactions with only the Z boson. This should have been expected. Neutrinos do not have electric charge l and therefore cannot have any tree-level interaction with the photon. The neutral current interaction of neutrinos is given by ..>f.(v,) nc
(15.35) The Feynman rule for the neutrino vertex with the Z-hoson is thus given by
Z"
(15.36)
lie
As for the electron, we ca.n rewrite the containing the electron field in Eq. (15.33) as ,,>(e) -_ 'xnc
1_ "L e(W 2"e"'( 9 ~ - 9 '8) iJ. 3
+ 9,-" F:"Y e 8 /i.
I
(15.37)
using L + R = 1. In of the gauge boson eigenstates, this can be written as 2
go
cas w
e-y"LeZ"+g'e'Y"e(-sinOwZ"+cosOwA")
2CO~ OW e-y" [L -
2
2sin Ow]eZ" + g' cosOwe'Y"eA" ' (15.38)
usmg the definition of the Weinberg angle from Eg. (15.21). The
338
Chapter 15. StaJldard electroweak theory
Feynman rule for the Z-interaction of the electron is thus given by
Z,.
e
(15.39)
e
The interaction betwccn the electron and A" agrees with QED if we identify the magnitude of the electron charge as e = g' cos Ow = gsinBw. ~
(15.40)
Exercise 15.5 Shorn that , for any fermion. oj charge eQ whose leftchiraL component has a T3 -eigenvaluf'. oj t3 and thf'. right-chirnl com· panent is a.n 5U(2) singlet, th.e. Fe.ynmnn rule lor the Z-inteTa.clion 1S
'~
co::> W
o
)'"(t3L _ Q sm 2 Ow).
(15.41)
Exercise 15.6 Show tha.t the ga.uge intera.ction~ oj th.c fermions uiola.te purity and C-imJnria.nce, but not .
15.3.2
Electron mass
So far. we have discussed the gauge covariant derivative involving the fermions, which include the kinetic term for free fermions and the gauge interactions. We now discuss the mass of the fermions. For any fermion field ,p. the mass term can be written as
(15.42) Take the electron field, for example. The eL transforms like part of a doublet under the SU(2) part of the gauge group, and the eR transforms like a singlet. The combination of the two cannot be gauge invariant.. Thus, in the fundamental Lagrangian of the standard model, there is no mass term for the electron. However, the Lagrangiall has spontaneolls symmetry breaking, and to drive this process, we had to introduce a doublet of Higgs bosans. Leptons can have gauge invariallt interactions involving the Higgs bosans. These arc Yukawa-typc interac.tions, given by YY"k = -h, (I!!,L¢e R + eR,ptl!!cl.) where he is
<\
coupling c.onstant.
(15.43)
15.3. Fermions in the theory
o
339
Exercise 15.7 Ch.eck that th~ intunction
te.rTni'l
of Eq. (15.43)
(lTe
inuaria.nt undt'.T the. gauge group SU(2)xU(1).
After spontaneous symmetry breaking, if we rewrite the Higgs field ¢> in of the quantum fields defined in Eq. (15.16), we obtain 2'Yuk
= -he
[;2
(eLeR
+ eNeLl + 'D,LeR¢> + + eRlIeL¢>-
+ ;2 (eeH + i "'Ys e()]
.
(15.44)
The involving the vacuum expectation value v of the Higgs field are the mass for the electron field, which have been generated by the spontaneous symmetry breaking process. Comparing with Eq. (15.42), we find th"t the electron mass is given in this model in of the Yukawa coupling constant he and the vacuum expectation value of the Higgs boson field: (15.45) The neutrino, on the other hand, remains ma.."islcss even after spontaneous symmetry breaking because of the absence of the lIe R.
15.3.3
Yukawa couplings
The Yukawa couplings, i.e., fields of the theory, appear using Eq. (15.45). For the gives the following Feynman
the couplings of fermions with scalar in Eq. (15.44). We can rewrite them unphysical charged sc"lar modes, this rules: -i J2m e R --~R v
-
'.j2Mw
' (15.46)
•
_ . J2m e L - -' ~me t
V-I
v'2Mw L
I
where in both cases we have used Eq. (15.19) in writing the latter form. Similarly, the neutral scalar modes have the following cou-
340
Chapter 15. Standard eJectroweak theory
plings:
>' > H
. me _
-1 V
-
e
e
.
. ---::-:,9<-".-_
-1.
2cosew
~
Mz '
(15.47) mc 11
_ '"15 -
[J
2 co", IJw
Tn iFz·
The neutrino has no coupling with the nelltra.l scalar modes since the model has no right-chiral neutrino.
15.3.4
Other fermions in the model
It is now known that there are three generations of leptons. In the standard model, the second and the third generations of leptons look exactly the same as the first one. The charged leptons in the second and the third generations arc the muon and the tau. We can write all the couplings involving these particles hy looking at what we have done for the electron and merely changing the electron mass to the relevant lepton ma..-c;s. There is also a neutrino corresponding to each charged lepton. If we try to incorporate qua.rks in the model, we find similarities and differences with leptons. Like lepton~. quarks also come in three generations. In anyone g:encration, the lcft-chiral components of quarks come in 5U(2) dOllblets and the right.-ehiral ones in singlets. The representations are 1
(2, 6)' UAll
dAR
2 (1'3)' 1 (1, -3)'
(15.48)
Here A is a generation index that fUllS frolll 1 to 3. Collectively, we will call all the UA'S a" u[>-type quarks and
the Yukawa couplings c.an connect any two generations. So for the quarks,
~Yuk
= -
L A.B
(hAuiiHdBR
+ h~BiiAI.J,llHR + h.c.),
(15.49)
15.3. Fermions in the theory
341
where = iTo'. We now see the differences with the leptonic sector, which we discuss in some detail. Here we have two kinds of . In the leptonic sector, we had only the first kind since there are no right handed neutrinos in the model. The right chiralities of the up-type quarks ,::ow give rise to the second term. In constructing this, we have used defined above, which contains the complex conjugates of the fields appearing in the multiplet <1>: 1 : (2'-2) o Exercise 15.8 Ij is
lln~ ~U(2)
(15.50)
doublet, ShOUl thllt ' does not
transform like a doublet, but ¢ = iT2· dOE'_l:I.
So far, there does not seem to be any problem. We can of course have Yukawa couplings involving two different fermions. The problem arises when we replace by the quantum fields using Eq. (15.16). This will now give mass as in Eq. (15.44), but they will involve two different fermions in general. We can, of course, partly avoid the problem by choosing the basis states of Eq. (15.48) in such a way that the 3 x 3 matrix h' is diagonal. This implies choosing a basis in which Ut, UO, 113 are really the physical particles, the up-quark 1l, the charm quark c and the top quark t. However, the matrix h need not be diagonal in this basis. This is a new feature, not encountered for leptons, since there was only one kind of coupling in the leptonic case which could always be taken as diagonal. In the present case, we obtain a mass matrix for the down-type quarks, given by (15.51) To obtain the eigenstates of the down-type quarks, we then need to diagonalize this mass matrix. This ca.n he donp. in general by llsing a bi-unitary transformation. It means that for any matrix m, it is possible to find two unitary matrices K and K' such that m = K'mKt is diagonal, with non-negative entries. These entries can then be interpreted as the masses of the physical down-type quarks, which we identify with the down quark d, the strange quark 5 and the bottom quark b. We will denote these by '?A with A = 1,2,3.
342
Chapter 15. Standard eJectroweak theory
The mass of these quarks coming from Eq. (15.49) can then be written as
L(KmK't)AB dALdBR. A.B
(15.52)
This shows that the eigenstates can be defined as {15.53) This has an interesting consequence on the gauge interactions of quarks. The interaction with the W-bosons can be written as in Eq. (15.33), where one must use the components of the same gauge multiplets. But since the physical down-type quarks are different from those in the multiplets, we get ,,,,(IV)
oLio<
=-
7
9 "(J2 '"AL'Y "d AL W+ "+ h .c. )
= - ~2 L
(UAL"Y"KAB9BLW: + h.c.)
(15.54)
A.B
using Eq. (15.53) to express everytbing in of the eigenstates. The matrix K is called the quark mixing matrix, or the CabibboKobayashi-Maskawa matrix. The Feynman rule for the W coupling for the quarks is therefore (15.55)
o
Exercise 15.9 Find the couplings of the Z bOSOT\s 'With the qUQTk eigensta.les. Show tl\o.t these do not inuolue the mixing ma.trix K I and i. giuen by Eq. (15.41).
15.4
Gauge boson decay
We have discussed processes involving photons in Ch. 9. Photons are massless, and hence cannot decay. In the electroweak theory, we have massive gauge bosons Wand Z, which can decay. This is a new feature of the electroweak theory, which we discuss now. To be specific, let us consider the decay of the Z-boson to a fermion-anti fermion pair. The coupling depends on which fermion
15.4, Gauge boson decay
343
we are talking about. In order to be general, let us write the coupling as 9
2int = - 2 cos 8W
-
h~(af - bns)! Z~,
(15.56)
where
(15,57) which follow from Eq, (15.41), The Feynman amplitude obtained from this would be
At = -
'98
2cos
W
,,(ph~(af - bns)v(p') (~(k),
where p and p' denote the momenta tifermion in the final state, and (~(k) the decaying Z-boson, We want the decay of unpolarized to average over the initial polarizations This gives
(15,58)
of the fermion and the anis the polarization vector of
Z bosons, For this we need and sum over the final spins.
(15,59) In writing this expression, we have neglected the rermion masses since the Z-boson is much heavier than all known fermions which it can decay to, Thus we obtain 1..... 1
2
9 (~(k)(~(k)) - cos28w x (aJ + bJ)(P"p'" + p"p'~ - 9~vp' p'),
-/, 2 _
(15.60)
There is also an antisymmetric contribution to the trace, but it does not contribute in the final count since the polarization average is symmetric in the Lorentz indices, as we will now see. Polarization sum In Ch, 8, we showed that the photon has only two physical degrees of freedom and calculated the polarization sum that can be used for
344
Chapter 15. Standard electIOweak theory
external photon legs in Eq. (8.88). For a massive gauge boson like the Z, this expression cannot be used for two reasons. First, k 2 is not zero for a massive on-shell particle with momentum k. Second, an arbitrary massive gauge boson need not couple to conserved currents. Since a spin-I boson callnot have more than three internal degrees of freedom, we can always choose the physical polarization vectors so that each of them satisfies the. relation (15.61)
and are normalized by f~(k)f.~(k)' =
-0".
(15.62)
In the case of the photon, only two such vectors are allowed for reasons described in Ch. 8. But in the case of a massive gauge boson, there can be three of them. For the 3-momentum k and energy E = Vk2 + M2, these are the two unit spatial vectors orthogonal to k, and a third one given by (15.63)
where k is the unit vector along k. This is usually called the longitudinal polarization vector since its spatial components are in the same direction as k, and so we have denote.d it with the subscript 'I'. The transverse polarization vectors, on the other hand, can be chosen just as for the photons. Combining, we find that the three physical polarization vectors satisfy the relation
(15.64)
To obtain the polarization average, we need to divide this by the number of polarization states, i.e., by 3. We see that the polarization average is symmetric in the Lorentz indices.
15.4. Gauge boson decay
345
Decay rate Going back to Eq. (15.60) now, we can write the matrix element square as
. (15.65) The rest frame of the Z-boson is defined by k"=(Mz,O).
(15.66)
In this frame, we can write
(15.67) where p = ~ M z since the masses of the fermions are being neglected. Using these, we obtain 1~12 = 3
2 '2 ') 20 (af+bj).
9 2MZ
cos
W
(15.68)
Plugging this into Eq. (7.119) and performing the angular integrations, we find the decay rate to be
qz --> f
f) =
g2Mz (02+b 2 ) 4811" cos 2 Ow f f
12sin 2
aMz (112 +b2 ). 2 Ow cos Ow f f
(15.69)
using Eq. (15.40) in the last step. Note thiit this is the decay rate into a specific pair of decay products. To find the total decay rate of the Z-boson, one needs to sum the rate.s over all possible final states. o Exercise 15.10 Using Mz
= 91 GeV. shaUl that the decay mte into a neulrino-antineutrino pair is about 165 MeV. Us€. sin 2 8 w = 0.23.
o Exercise 15.11 Using Mw
= BOGeV, ShOUl that the mtc lor the decay W+ -. e+lIe is abou.t 220 MeV. You CUll negLect the mass of the. electron.
346
Chapter 15. Standard electroweak theory
o Exercise 15.12 Con8ideT the decay of the Higg8 bo80n H into" lepton-"ntilepton pniT t+ t- (where t = e-, /'- etc). Show th"l lhe decay rote. i8 gi"en by
r(H -t+r) =
<>m)MH
8sin 2 Bwcos2 8wMi
(1 _
4m))~' M'h
(15.70)
IHint: Ha\le you sun this done ea,,-lieT in th.e book in 80me jonn?1
15.5
Scattering processes
15.5.1
Forward-backward asymmetry
In §9.6, we discussed lhe process e- e+ -. /l- /l+ in QED. We can now calculate how the cross section is modified by weak corrections.
"(, Z
H,(
Figure 15.2: Lowest order dioagrams (or the process e-e+ -1J-J.1.+ in the electrowea k theory.
There are four diagrams in the lowest order, as shown in Fig. 15.2. Of these, we have already calculated the amplitude for the photon mediated diagram in §9.6 and found (15.71) where 8 is the Mandelstam variable. Notice also that the diagrams mediated by the Hand ( contain couplings which are proportional to the fermion masses. It was already argued in §9.6 that it is perfectly justifiable to neglect the electron mass in studying this process. Hence these two diagrams do not contribute to the amplitude in this approximation. The coupling of the electron and the muon with the Z-boson are the same, given in Eq. (15.39). Using Eq. (15.40), we
15.5. Scattering processes
347
can write the Feynman rule in the form 2e
-
. 28
sm
w
'Y,,(gV - gA'YS) ,
(15.72)
where (15.73) Then we can write the Z-mediated amplitude as 2
e .fz Jltz = - [jj(k2h~(gv - gA'Y5)U(kt} ]
x
by choosing the
~
[u(ptl'Y~(9V - 9A'YS)V(P2)] ,
(15.74)
= 1 gauge, and defining
fz =
sin 2 28
•
w (s -
M~)
.
(15.75)
So the square of the total amplitude, averaged over initial spins and summed over final ones, is given by
IJltl 2= ~ L
IJIt, + JltZ!2 .
(15.76)
SP1~
There are four , one of which has been calculated in §9.6 and found to be (15.77) where now we neglect the mnon mass as well, anticipating that the corrections from the Z-boson can be important only if s is comparable to M~, and therefore much higher than m~. The square of the Z-mediated amplitude is given by
IJltzl2 = e4fH(g~ + g~)2(1 + cos 20) + 8g~g~cosOl.
(15.78)
As for the cross , we find
JIt;Jltz = e:{ZTr
[jln~(gv -
9A'YoJjlnP]
xTr [Pl'Y~(gV - 9A'YS)P2'YP] .
(15.79)
Chapter 15. Standard electroweak theory
348
The 9A9Y vanish because they multiply one trace involving "15 which is antisymmetric in the Lorentz indices A, p, and another trace which is symmetric in these indices. We are then left with
.41;.41.
+ h.c.
=
2e4fZ[9~(l + cos 2 0) + 29~ cosO]. (15.80)
Adding up all the contributions, we can write
(15.81) where
at = 2Jz9~ 2 a2 = 4 f Z9A
+ fi(9~ + 9~f , 2 2 + 8f2Z9Y9A'
(15.82)
The at contriblltion has the same anglllar dependence as the pure QED result. The important qualitative differellce from the pure QED result is the appearance of the a2-term. which is proportional to cos O. Without this term, the differential cross section would be invariant under the trallsformation (j -+ 1f - e. In other words, it would have been forward-backward symmetric. The a2-tcrm introduces a forward-backward asymmetry. The forward-backw"rd asymmetry is ill general defined by the ratio
.A
"/2 dC!;" dC! dOsinO- dOsinOu dn "/2 dn
-_ 1
r
In
dC!
(15.83)
dOsinO dn
with the integration over the azimuthal alll4lc implied ror each illtegral. For this process. Eq. (15.81) gives
A=~
0'2
8 1 + al
(1584)
o Exercise 15.13 Eunluf'ltc the amount oj JOl"wurd-buck1..llurd u.81Jm. mctTy in lhil:l proC('.~~ ul ~ = IlHsumi.ll!1 sin 2 Ow = 0.23.
tA-11,
349
15.5. Scattering processes 1I,,(k) Z
e(p)
e(p')
Lowest order diagram for the process v~e electroweak theory.
Figure 15.3:
15.5.2
--+
vpe
in the
Low energy weak interactions
The process we discussed above can be mediated by electromagnetic as well as by weak interaction. Now we consider processes which can go only via the mediation of the W or the Z. These will be purely weak processes. As an example, we take the elastic scattpring of muon-neutrinos (II,,) by electrons. This can be mediated by the Z-boson exchange, as shown in Fig. 15.3. There is no diagram with neutral scalar exchange because, as we saw in §15.3, the neutrinos do not couple to neutral scalars.
The Feynman amplitude of this proceso call he directly written using the Feynman rules given in Eqs. (15.36) and (15.39):
.
,.J{' =
Ap (i )2 _ig 9 -
2cosOw
2
2
q - Mz
x [U(k')-YALu(k)] [u(p')-yp(9V - 9A'Y.;)U(pJ] ,
(15.85)
where we have used the Feynman- 't Houft gaUR" and introduced the notation q = k - k' = pi - p.
Let us suppose now that the energies involved in the process are much smaller than M z. Then q2 « M~. and we call neglect the momentum dependence of the Z propagator to write .4£ = ( 2M z
~osew
r
x [U(k')-YALlL(kJ] [u(p')-yA('I\' - '1nr,}1L(f/)].
(15.86)
But this is exactly the n.lnplitude that we would have' ohtained if we
350
Chapter 15. Standard eJeetroweak theory
had started from a 4-fermion interaction Lagrangian
2lnt =
~ [,p(""n~(1 -
"YS)V>hl]
[,p(,)"Y~CqV
- 9A"YS),pk)] . (15.87)
provided we had identified the Fermi constant as g2
-
g2
- 4v'2M~ eos2 Ow - 4v'2Ma, ,
(15.88)
using Eq. (15.22) in the last step. Thus at low energies, weak interaction is given by 4-fermion interactions. In this limit therefore, the calculation of the cross section is very similar to what was shown in §7.5. We omit the details and write directly
1A'l12 =
16C}[(gV
+ gA)2(k. p)2 + (gV
_ gA)2(k' . p)2
-(g~ - g~)m;k. k'] (15.89) for this case. Then, in the Lab frame in which the initial electron is at rest, the differential cross section can be written down using Eq. (7.118):
da dO
C2p m 2,
1
4,,2
(m, +w -wcasO)2
x [(9V
+ 9A)2w2 + (gV
- gA)2 w'2 - (9~ - g~)ww'(1 - cosO)].
(15.90)
Here 0 is the angle between the initial and the final neutrino momen ta, and
(15.91)
is the energy of the final neutrino. The total cross section can be found by integrating over the angular variables. For w » m, one obtains a =
C}m.w [
2,,'
(gv
1 + gAl 2+ 3(gV
o Exercise 15.14 The, muon deco.'Y, tlin u W -excha.nge gra.ph.
jJ.-
----1
- gAl e-
2] .
+ Vc + V/-l.'
(15.92) VTOCe.eds
Draw the diagram and write down the amplitude. NoUJ URI'. Eq. (15.88) to show that ij th.e energies a.re ~ma.ll C'.ompurtd to the ",' ~m(l~~, the tOHlf'.Rt order amplitude is the sa.me. flR that obta.ined Irom the 4-jermiort Lagra.ngian of Eq. (7.36).
15.5. Scattering processes
o
351
Exercise 15.15 For the elastic 8Cilttering between the Lie Q,nd the electron, there is a1\ additional diagram with a. W -bo80n exchange which contributea at this order. Add th.e a.mplitude of this diagmm with that oj the Z-medio.ted diagrom, a.nd show that IAfj 2 is giuen b-y the same form 0.8 in Eq. (15.89) prouide.d we define
1 2sm · ' 8w, 9V=2+
15.5.3
9A =
1
'2.
(15.93)
High energy scattering
As a last example of scattering processes in the standard electroweak theory, we consider the pair creation of W bosnns: (15.94)
The initial center of mass energy has to be high enough to produce the W's in the final state. Therefore we can neglect the electron mass.
1, Z
(a)
(b)
Figure 15.4: Lowest order diagrams for e-e+ - W-W+.
There are five diagrams for this process at the tree level. One of them, shown in Fig. 15.4a, has an intermediate neutrino line. Two others are collectively shown in Fig. 15Ab, where the internal line can be either a photon or a Z-boson. In addition, there are diagrams involving exchanges of neutral scalars Hand (. The calculation is very lengthy for unpolarized particles in the initial as well as in the final states. So we will demonstrate this calculation in a very specific easel characterized by the following conditions: 1. The electrons have positive helicity.
Chapter 15. Standard electroweak theory
352
2. Only the production of longitudinal polarization states of the W's is considered.
3. The center of mass energy is much higher than Mw, i.e.,
Ma.
S»
Since we are neglecting the electron mass, the diagrams involving neutral scalar exchange vanish. In addit.ion, the first assumption means that the electrons arc right chira!' In this case, Fig. 15.4a does not contribute either. This can be underst.ood from the coupling given in Eq. (15.34), which shows t.hat it. vanishes for a right-chiral electron. Since the couplings involve a ,,-matrix, it also implies that the positron h"" to he right-chiral as wei!. The second assumption means that the polarization vector for the final state particles can he chosen as in Eq. (15.63). Moreover, since E = k due to the third "",,umption, we can write the longitudinal polarization vector simply a.s
(15.95) These considerations allow us to write down the amplitudes in the Feynman-'t Hooft gauge ""
where f± are the longitudinal polarization vectors for the W±, and V~v~(L, k+) =
(L -
k+hg~v
+ (2k+ + L)".'!v; - (2L + k+)vq~~. (15.97)
o
Exercise 15.16 C:u: the FCllumnn rule for cuhic coupling from Fi{J. 14.1 and the. fLt'.ji.nition:; oj UH~ Z boson lLlut th.e. photon in. Eq. (15.20) to 1Je.rif1J lhf' ~VWZ una WH/A t·oll.phn.~p! UlH'rt (~houc.
Since un = Rll where R = ~(I + ",,), th,' kft-d,ira! project.or in the Z-couphng docs not contribute. TheIl. using Eq. (15.10). we can
353
15.5. Scattering processes combine the two amplitudes to write
0-S_lM1)
2 vf(= vf(,+vf(z = e [VR(p+h'UR(P_)]
x V~",(L,k+),~(L)<~(k+).
(15.98)
H we now use Eq. (15.95), we obtain 1
2
+ Mw)(k+
M;t (k+· L s
M2 (k+ - L)"
(15.99)
2 w llsing s
»
Mev in the la.~t step to write k+ . k_
- L),
= !s in the center of
mass frame. The factor in Eq. (15.98) coming from the propagators can also be simplified as 1 s
1
s- M~
M1
M1 .
---;-(--"M:-:e,C7 '" - -s2 ss- z)
Putting all of these together, we finally obtain the
(15.100)
~mplitude to
be
(15.101) using the relation b
o
Exercise 15.17 * Show that one obtains c:mctllJ th.e same a.mplitude as in Eq. (t5.101) if the external Ht'±-linea aTe imagined to be repln('.ed by Ute ¢± tines. You need to deriue titf'. Fe:ynman ruLes jar the 4> coupLings to cnleulate this dia.grum. INote: Thi.~ cqui\lUlence of the amplitude holds only for the lonfjitudinul polnrizn.tion stale a.t energies much high.er than thf'. gClU!Je hOMn tnc eqtLivalenCf~ theorem. I
In.U~l(,
and is known as
There is now no average over initial spin ~illce we have taken the initial fermions in specific spin states. So wp cannot directly use the spin sum formulas while calculatinR the ;jquare of this matrix element. Instead, we use the chirality projec.tic)J] operat.ors to write
(15.102)
354
Chapter 15. Standard eJectroweak theory
It then follows that
[VR(P+ hPUR(P-)] t [VR(P+ h>'UR(P-)]
= [u(p_hPRv(p+)] [v(p+)-Y>-Ru(p_)]
L
=
[u(p_ h P Rv(p+)] [v(p+ h>' Ru(p_) 1.
(15.103)
spin
The last equality works because of the projector appearing in the bilinears. Even if we sum over all spins, the projector will pick out only the corresponding projection. This allows us to use the spin sum technique to write the above expression as
Tr [p+ -Y>-Rp_ "'(PRJ = 2(p~P".
+ p~rI;.
- g>'Pp+. p_)
+ <-term. (15.104)
The quantity denoted by IE_term' contains tl1e antisymmetric tensor, and is irrelevant since the expression obtained here is contracted with something which is symmetric in >., p. So we get
1.41'1
2
e
4 (
>.
-P
= 28 2 cos4 Ow P+I'-
>. p i > . ) + p-p+ - 28g P (k+
- L)>.(k+ - k_)p
e4
sin 2 O. (15.105) 4cos 4 Ow Here 0 is the scattering angle, which we have taken to be the angle between p_ and k_. We have also used kinematical relations like p+ . p_ = k+ . k_ = ~8 which are valid with our assumptions. The scattering cross section can now easily be obtained using Eq. (7.106): 7,",,2
a = 68 cos4 Ow .
(15.106)
o Exercise 15.18 Consider the decay oj muons at Test with. spins aligned aLong the direction s. With the interuction of Eq. (7.36) 2 and using spin 'Projection operators, ShO'Ul that l.lr1 is obtained by replacing p>' b'Y p>" - Ullin>' in Eq. (7.49L where n>' is defined by Eq. (4.81). Assuming me = OJ ShOUl that thf': differential aecay is giuen by
(15.107)
where 8 is the angle between s and the, electron momentum. {N ote: Under 'Parity 8 -I' IT - B since the spin remains invariant while the momenta are reuensed. The cos B term then 8ignals parity uiotation·1
15.6. Propagator for unstable particles
15.6
355
Propagator for unstable particles
Clearly, the scattering formulas derived in §15.5 do not hold if the incident particle energies are such that s = Mi. For example, the quantity !z, defined in Eq. (15.75), blows up. This is because the Z-boson is an unstable particle. In writing the propagator of any unstable particle, we must take its non-zero decay width into . To see how this can be done, let us consider a scalar field for the sake of simplicity. For a stable particle, the solutions of the free equation of motion are plane waves. If however the particle has a decay width r in its rest frame, the free particle solutions "'(x) should decay with time as (15.108) up to normahzation factors. The factor /H/ E occurs in a general frame to for the time dilation factor for its lifetime l/r. Thus we can write
"'(x)
~ cxp (-iEt + ip' x
-
;~rt)
.
(15.109)
In order to identify a particle by its energy, the uncertainty in energy measurements must be much less than its energy, i.e.\ .6.E « E. In the re.st frame of the particle, this implies l:J.E «m. Due to the time-energy uncertainty relation, the time taken to make the measurement will satisfy l:J.t » l/m. Unless this time is much less than the lifetime l/r of the particle, no such measurement would be possible. Thus for a particle interpretation of the quanta of the field, it is necessary that the condition r « m and therefore r « E be satisfied. In that case, ignoring O(r' /m') corrections, the equation of motion for the field "'(x) can be writteu from Eq. (15.109) as
(0 + Tn' - imr) "'(x) = O.
(15.110)
We can now use the methods of §3.7 to fiud the propagator of this field. We will get l:J.p(p) =
I
p' -
.' Tn' + unr
(15.1ll)
Chapter 15. Standard electroweak theory
356
A similar analysis for vector bosons would give the following propagator for an unstable vector boson in the Feynman· 't HODft gauge: D"v(p) = -
2 g"v. . p - rn 2 + Mar
(15.112)
Since r « Tn by OUT as~umptjon, the extra term added is immaterial unless p2 is very close to ,n'l, i.c., unless the particle is on-shell, or very nearly so. But for a nearly on-shell particle in the intermediate state, one should use this propltgator.
o
Exercise 15.19 Find thf", jorwnrd-backulO.nl. o.slJmme.trll in e- e+ Ali. Take sin 2 8w = 0.23 UR hf'jore, and r z = 2.49 Ge V.
p.- Ji+ o.t s =
15.7
Global symmetries of the model
The standard clectroweak model is ba.'.)cd 011 a. local symmetry with the gauge group SU(2) x U(I). Thi. of r.our.c entails the same global symmetry. But in addition, the ,tandard model has some other glohnl symmetries as well. One of these is the ba.ryon number symllletry, which is defined ~ (15.113)
where q herc stands for all quark fields, and B i. the baryon number of quarks, usually taken to be Antiquarks havc B = All other particles are assumed to have zero baryon number and hence do not transform under this symmetry operation. One can go back and check easily that no iuteraction of the standard model changes a quark into a lepton, hence this symmetry. In the leptonic sector also there are symmetries of this kind. For example consider
1.
-1.
1
(15.114) where e is any negatively charged lepton or it.s corresponding neutrino. Conventionally, the quantity (" called it'plon number, is taken to be 1 for all of them and -I for their l1.nt.iparticlco. All other particles have zero lepton numher. In fact. unlike the quarks, there is. no intergcnerational mixing in the ltollic spctor of the :::;tundard model, so even the lepton numbers corrcspondiug t.o each generation
15.7. Global symmetries of the model
357
is conserved. In other words, if we assign a global quantum number I to the electron and the electron-neutrino, -I for their antiparticles, and zero to all other particles, the quantum number will be conserved. The same applies if we assign the number to any other charged lepton and its associated neutrino only. These symmetries have a very different character than the gauge symmetry. First of all, these are global symmetries, as opposed to the gauge symmetry which is local. Secondly, these are not symmetries that we imposed on the model. We built up the theory with the gauge symmetry and with the specific fermion content, and after doing that we found these extra symmetries in the theory. Such symmetries are called accidental symmetries. There is another difference. In this book, we have always relied on perturbation theory to derive the interactions. However, if nonperturbative effects are allowed, it can be shown that a global U(I) quantum number '§ cannot be a symmetry of the model unless it satisfies the relations
The two sums in this equation are over the left-chiral and the rightchiral fermions, and !Z is any diagonal local charge. The general condition is somewhat more involved, but this is enough to show that baryon number and lepton number are not symmetries if we include non-perturbative effects. However, the difference B - £. is still a global symmetry in the model, sinc.e it satisfies all the tests like the ones given in Eq. (15.115).
o
Exercise 15.20 Verih th"t
(~=-~) (B-£f2 = O. (~=- ~) (B- £)..'11' =
0,
where. 12 co.n be the dectric chnrge. or liu'. h'Yperchuige Y. tha.t each quul'k is uctua.U'Y Cl C:010T triplelo[
(15.116) IHint:
Appendix A
Useful formulas A.I
Representation of ')'-matrices
The 'Y-matrices are defined by the anticommutation relation (A.I) and their hermiticity properties: ('Y~)t = 'Y0'Y~'Y0.
(A.2)
We mentioned tbat these two properties do not uniquely define the specific form of the matrices. In particular, if two sets of matrices h~) and h~} both satisfy Eqs. (A.I) and (A.2), they will be related by
(A.3) for some unitary matrix U. In view of this fact, we tried to do everything in a way that is manifestly independent of the representation of the 'Y-matrices. In order to gain some insight into the solutions, sometimes a specific representation is helpful. For example, in §4.3.2, we used the Dirac-Pauli representation, in which the matrices are given by 'Y
0=
(I 0) 0 -I
. (0
"'(I =
0")
_a i 0
(A.4)
'
where I is the 2 x 2 unit matrix and the O"'s are the Pauli matrices: 1
(0 I)
0'=10'
0'
359
3
=
(I 0) 0 -I
.
(A.5)
360
Appendix A. Useful formulas
The advantage of this representation is understood by looking at the solutions for the spinors derived in §4.3.2. For a non-relativistic particle, the lower two components of then-spinors are negligible. Hence the non-relativistic spinor is effectively a two-component one, which is easier to use. Thus the Dirac-Pauli representation is particularly helpful for the non-relativistic reduction of a problem. It was used for this purpose in §11.2. In this representation, "15 and o~v take the forms
~5 = (JooJ), I
0°' = i (0°,. 00') ,
aij :=::: [ijk
(ok0 °) uk
I
(A.6)
where "'jk is completely antisymmetric symbol with "123 = + 1. Another useful representation is called the chiral representation, in which
°
° -J)
(A.7)
"I = ( -J 0
Notice that the matrices "I' have the same form as in the Dirac-Pauli is different. In this representation, representation, but
,0
(A.B) Thus the chirality projection operators in this representation have the form
~(l+"I5)= (;~),
(A.9)
So the chiral eigenstates will have zeros for either the upper two components, or the lower two. If one has to deal with chiral eigenstates explicitly, this is a more convenient representation to use. It is used extensively, for example, in discussing supersymmetric field theories, a subject which is beyond the scope of the present book. In this representation, Oij arc the same as in the Dirac-Pauli representation, whereas _ . a 0, 1
(0' ° ) 0
-0
'
.
(A.IO)
A.2. Traces of 'Y-matrices
361
A third useful representation Here,
IS
the Majorana representation.
;u
1
'Y = (
3
o
-iU
o
(
0 )
iu 3 1
0
_;u 1
)
(A.Il)
from which we can calculate 2
02) 'Y5 = ( uO -u
.
(A.12)
Notice that all the 'Y-matrices are purely imaginary. For one thing, it assures that the u~v's are also purely imaginary. In the Dirac equation, the operator i-y~a~ - m is now completely real. As a result, if we have a solution where all the components are real at one time, they will remain real for all subsequent times. This is very useful if there are fermionic particles which are antiparticles of themselves. o Exercise A.I Find
A.2
(1~./I18
in. the Mo.joro.na representation.
Traces of ,,-matrices
For the sake of notational simplicity, we will use a number of 4vectors which we will denote by a" with r = 1,2,3· , " and consider traces of the form
(A.13) We first show that Tn vanishes if n is odd. For this, we use the properties of the matrix 'Y5. Since its square is the unit matrix, we can write Tn(al,02,'''On) = Tr ['Y5'Y5~1~2'''~nJ = Tr 1'Y5~1~2'" ~n'Y5J ,
(A.14)
using the cyclic property of traces. We now take the 'Y5 appearing at the end through the ~k 's. Since 'Y5 anticommutes with all the 'Y" 's, we will obtain a minus sign each time we go through one of the slashed vectors. To go through n of them, we will obtain a factor
362
Appendix A, Useful formulas
(_I)n, at which point the two factors of ", will be sitting next to each other and we will forget them since tilo:;e two factors will give 1
the unit matrix. Thus we obtnin
Tn = (-I) nTn '
(AI5)
This equation says nothing if n is even, but if n is odd, it proves that
Tn
vanishes.
The non-trivial traces thus involve an p.vetl 1l1llnber of I-matrices. For the ease of n = 2, we obtained the trace in ~7,2,1 at some length. Here we repeat tile derivation ill a. different Hotation, which will be lIseful for the trac.es with higher number of l'-matrices. For any two
vectors a ano b,
1# = 2a b - N,
(A.16)
which follows from the anticommutation r<'iation of the ,'-matrices, Therefore,
T1 (a"a2) = Tr 1~'~2] = Tr 12a, 'a11 - Tr [~2~,1 = Tr [2a, . a2]- Tr [~,hJ where we have lIsed the c.yclic property of traces. , we obtain
T 2(01,a2) = Tr [a,· a2] =
'
(A.17)
By rearranging the
0, '"1Tr [IJ,
(A.18)
where we have explicitly written the unit matrix implied in Eq. (A.16). The trace of the IInit matrix is 4, so filially we get
T2 = 4o, . "1.
(A.19)
Once this is done, all higher traces can 1)(> expressed in of T2 . For example. starting now with the definit.ion of T, and \Ising this relation repeatedly, we obtain
T,(a" a2, a3, a,) = Tr [~'~2~3~,J = Tr [(2a, . a2 - ~2~tl~3~,J = 2o, . 02 T 2(a3, a,) - Tr [~2~,h~.d = 2al . a2 T2 (03. a,) - Tr 1~2(2a, . n:j - ~3"1 )~'1J = 2al' a2 T 2 (a3.(/.,) - 2al' "3 T2("2,''-l) + Tr [~2~3~1~,J = 2a,· a2 'f.,(a3' a,) - 2"1' a3 T2(1t2."") +2a, . a., T,(02, a:l) - Tr l~d:l~,~,1 . (A.20)
A.2. nares of "I-matrices
363
The remaining trace is equal to the original trace due to the cyclic property of traces. So we obtain
T4 (01,02,03,0.)
= 0, . 02 T2(03, 0.) - 0, . 03 T2(a2. a,) + 0, . a4 T2(a2,a3) = 4(a, . a2"3' a, - a,· a3a2' a. + a, . "4"2' a,,) (A.21) In a similar way, we can show that
T2n (al
l
a2,···a2n)
= at· a 2 T2n-2(a3,Q4,···a2n) -al . a3 T2n -
2 (a2, f11,'"
a2n)
+al' a4 T2n -2(a2! a:t,Us ... a:l 71 )
+ ... (A.22)
The traces can also be expressed in their bare form, without using the 4-vcctors. E'or ex;\mple, from Eq. (A.19). we c;\n extract the coefficients of afa2 from both sidC'B"i to write (A.23)
Simil;\rly. from Eq. (A.21), we can take t.h,' c(>-efficient" of "'t"2a5a~ to obtain (A.24) Since "(5 contains four ,,-matrices, traces involving 1'.") can bp. nOI1vallishing only if there is atl even Humber of ,-matric~s lIlultiplying 1'5. If this numht:'r happens to be zero, tlH' lracl' is zero, since we ha.ve shown ill eh. 4 that 15 is tra<:eless. The trace also vanishes if 1'5 is ar.eolll panicd by two 1- matrices. Til is is hecause
Tr 1"1,,)""15) - - Tr 1"1",-,',,,] - - Tr h,,"1,,),,].
(A 25)
At the fir~t equality sign, we lH1ve used the a.llticOllllllllt.ing: property of 151 and at the sc<.~ond, we have llsed t.IH' ('.velie propert.y of traces. \Vhat \...·e have outaincd iti that the trace sholllJ be alltisyrnmctric in the indices IL, v. There is no sHch rnllk-2 t.ensor in the structure of space-time. and hence tlH' tra.ce must vanish. Tlw smallest non-vanishing trace involving "/.5 thlls cOllla.ills fonr lI10re l'-lIlat.rkcs.
364
Appendix A. Useful formulas
These four have to be all different, because if any two are the same, the trace can be reduced to the form in Eq. (A.25). Because difference "I-matrices anticommute, the trace has to be proportional to the antisymmetric tensor. The proportionality constant can be fixed by calculating Tr hO"lI1'2'Y3'Y51, (A.26) To obtain higher traces involving "15, it is useful to use the identity "Ip"l""I' = gp""I' - gp'''I"
+ g",gp -
i
(A.27)
which can be checked by substitution. One interesting property of the "I-matrix traces can be proved using the matrix C introduced in §1O.3. This is
Tr [~Hj2'" ~2n-l~2n] = Tr [~2n~2n-l .. '~2~11
(A.28)
o
Exercise A.2 Proue Eq. (A.28).
o
Exercise A.3 ShOUl that a.n" ma.tN 'Which ant\commutes with all the ~JJ mU8t be a. multiple oj "(5. [Hint: Use the ba.sis matrices gi'Uen
in Eq. (.\.14).1
A.3
The antisymmetric tensor
The antisymmetric, or the Levi-Civita tensor\ is a rank-4 tensor which is antisymmetric in the interchange of any pair of indices. Thus, the tensor can have non-zero components only when all the four indices are different, and in this case, if we define its value for any order of the indices, the antisymmetry fixes the values for all others. We have chosen '0123
=
+1.
(A.29)
Following the usual rules for raising and lowering of indices, this would also imply ,0123
= -1.
(A.30)
When there is a pair of antisymmetric tensors, we can express them in of the metric tensor:
(A.31)
A.4. Useful ;ntegration formulas
365
where the square brackets indicate that all permutations of the lower indices are to be added, even permutations with a factor of + 1 and odd permutations with a factor of -1. This can be verified by direct substitution. When one index is contracted, we obtain f
IJ-V>'P
E~II'),.!p'
(A.32) We can now go further and contract one more index. This will give
(A.33) Another contraction would lead to the identity ~v>..p
E
fiJv>.(I -
-
69pl' p
(A.34)
and finally, if all the indices are contracted, we obtain f iJV>'P fp.v>.p
24 ,
= -
(A.35)
A.4
Useful integration formulas
A.4.1
Angular integrations in N-dimensional space
Consider an N-dimem:iional space with Cartesian co-ordinates XI,X2···XN. Let us denote the magnitude ofx by r, i.e., N
LX; = r
2
.
(A.36)
i=l
If we integrate a function of T over the entire space, we should be able to express the integral in the form
J
d N x f(T) = eN
faOC dr TN- 1f(T)
(A.37)
for some number eN. Essentially, this eN b obtained by integrating over the angular variables in a spherical co-ordinate system. Our ta.c;k is to determine this eN.
366
Appendix A. Useful formulas
Since the angular integrations do not depend on the form of the function f(rL we can use any convenient function to obtain eN. Let us consider
(A.38)
fir) = exp (_r2/a2) .
In this case, the integral appearing on the right side of Eq. (A.37) can be rewritten as
fo~ drr N - 1 exp(_ r2 / a2 )
=a N fo~ dppN-1
In mathematical analysis, the is defined by the relation
r -function
cxp(-/). (A.39)
for 'lny complex variable z
(AAO) where we have written (
=
p2 in the la.~t step. Thus we ohtain the
result:
Right hand side of Eq. (A.37) =
~CN(/"f(NI2).
(AAl)
On the other hand. using Eq. (A.36), we can write the left side in the form
JdNxex[l(-~x:la2)
= [[:dTex p =
(-x 2/ a2
aN 1r(1/2)]N .
)r
(AA2)
For N = 2, Eqs. (AAl) and (AA2) give f(1/2) =.fii. For arbitrary N, we then obtain
eN o
27r N /'l = r(NI2) .
(AA3)
Exercise A.4 IJ u:t: considererl
f(r)={~
ifr
~
if,. >
R,
n.
(A4~)
the integTation clt'.jill('.(l in Eq. (A.3"1) stw'ld,d. hl1.n. gi,llf'lt 1I~ tlH' votume of n sphere in N~di1rl('1L:flon. UI\f> the Jormtlln for (',\,' !)il:ell. ClhOl~f' tn ucrifll thllt. one ohtlLi.ll~ the ('orn'cl "uol'U.1lL(," for N = 2.3.
A.4. Useful integration formulas
A.4.2
367
Momentum integration in loops
In loop diagrams, after introducing the Feynman parameters, the momentum integrations can be reduced to the form , _JdNk (k')' I. (a, N) (21f)N (k' _ a2 )•.
(A.45)
Here, N is the dimension of space-time. For convergent integrals}
it can be taken as 4. In order to tackle divergent integrals as well, a subject which has been discussed in Ch. 12, we keep N arbitrary here. After performing the Wick rotation, the integral becomes '( N) 1,a,
.(
=,
-I
)'H
J
N
d kE (k'jy (21f)N (k~ + a2)' '
(A.46)
At this point, we can use the result obtained above for the integration over the angular variables to write (A.47) where T
roo
_
Jo
J,(N) -
y"-l+~N
dy (y + I)'
(A.48)
with y = k1/a'. This remaining integral can be expressed in a standard form if we substitute
z=
1
y+1
(A.49)
In that case,
Y I - z = --"-..,y
+1'
dy dZ=-(y+I)2'
(A.50)
So we can write
J;(N)
= 10"0 (Y~YI)2 =
fal dz
(1 -
(y~J-l+jN (y~J-'-l-jN
zr-l+ jN Z'-'-I-jN
=B(r+~N,s-r-~N),
(A.51)
368
Appendix A. Useful formulas
In the last step, we have introduced the beta-function of mathematical analysis, which can be given in of the f-function defined above:
B(
z,
z') = f(z)f(z')
(A.52)
r(z+z').
Substituting this into Eq. (A.47), we finally obtain
i(-1) r +o -..,._'---;-;;f(r + !N)f(s - r - !N) ( ) I. (a, N) = (41T)N(2 (a 2 ).-r-1 N f(~N)f(s) . A.53 r
For convergent integrals, we can put N suppress the dimension. This gives 1 (a2).
="
f(r r
2
4 as mentioned earlier and
+ 2)fts f(s)
r - 2)
(A.54)
Appendix B
Answers to selected exercises Ex. 1.3. N 1 "(ala. + -)"",. 2 .,
H = L-
Hamiltonian :
I
t
i=l
state : N
number operator:
.%=
Lara.. i=l
Ex. 1.1. Maxwell equations are given in Ch. 8. The Lorentz force law on a particle with charge q is dpl' = qF~. dx•. dr dr
where
x~
are the coordinates of the worldline of the particle, and r is proper time, dr'l = g",.udx""dx v .
Ex. 1.8. r = 2 x lO-6 s . 8V Ex. £.£. (0 + m 2 )¢ = - ()¢. 8V Ex. £.3. (0 + m 2 )¢' = - ()¢ .
Ex.
£.~
• The answer appears in Ch. 8.
369
370
Answers to selected exerdses
Ex. fl.7.
= Iq•.
a) P.
00
b)
H= ;1 L
(p~ + z2w~q~).
k=1
c) 1£1•• a!n] d) H
= o'm , 1£1., amJ- = [al, a!nl- = o.
~ -2-(£1'£1. """. 1 + £1.£1.). 1 = L. k=l
Ex. 2.8. a) T"V = 8"8" - 9"v!f, b) T"" = 8"18" + 8"8"1 - 9""!f. c) TIJI/ = _F~P8tJ A p _ gJ.JII,:LJ. Ex. fl.9. a) j" = ;q(18" - 8"1). b) j" = iq(18" -
8"1 + 2iqA"I-x"!f. Ex. 3.5. P" Ex. 3.7. Q
J J
=~
= iq
d3 p P" (£1 1(p)a(p) +a(p)af(p»). 3
d p H(p)a,(p) -
Ex. 4.8. mv,(p)-y"v,(p) Ex.
4.9.
a~(p)adp)].
= -p"v,(p)v,(p).
v(p'h" v(p) = - ,:. v(p') [(p + p')" -
i""" qvl v(p) .
Ex. 4.1fl. Same as in Eq. (4.61) and Eq. (4.62).
Ex.
4.26e >.
.,;f(AJlIJ -. >.
!1/J'Y>'aJ.tIll/J
-
(T\x u
T\,xj.J),
-
where
>.
T " = 1/"'Y 0"'" - 9 "!f. Ex. 6.2. Ex.
G" v'2 "'(".)'1"(1 - '15)"'(.) "'(n)'1"(1 -
'15)"'(p) .
6.4. S(3') = (_ih)3
/-
x
(2rr)'o'(k _ p _ p') [1
J(~:~,
1
1]
v'2WkV )2 Ep V .j2Ep ,v
i6p(q) [u.(p)iSF(p - q) iSF(p)V,·(p')1 '
Answers to selected exercises Sf(3,'")
= (
-,
(2
h)' x
"
)'6'(k
J(~:~,
371
-1'-1'
')
[1 )
2W k
1
I]
V J2E p V J2E p'V
iilF(q) lu.(p)iSF(I') iSp(p - q)v.,(p')] .
Ex. 7.3. Yes. Ex. 7.5.
Ex. 7.6 •
r=
G'FffiJ's
r(1f+ .- e+ve ) r(7r+
8:~ +)
192,,3 (I -
m' mP
• =2
J.L+v j , )
----<+
, (mm;.--m')' m; . 2
Ex. 7.9. ]] GoVapproximately. Ex. 8.1 • 01,FIW
+ A,f2 All
Ex. 8.2. B(x) = Bo(x)
+
=
jV.
J
d'x' Go(x - x')f(x'), where Go is the mass-
less scalar propagator (Green's function.for the wave equation) and 80 satisfies the wave equation.
Ex. 8.5. Physical states: a!(k) 10) for,. ~ 1,2. Ex. 8.6. :PI': =
J
d3 k k P
L, a;(k)a,(k), wherp kO =
Wk·
This expres-
sion is valid on physical states selected by the Gupta-Bleuler condition, so that only the transverse modes contrihute for cac:h kf' in the integral. (Physical quantities like Hamiltonian and momentum are independent of ~ when restricted to physical states.)
Ex. 11 ..1. eF(O).
Ex. 12.3. Define EJ.'olvt3 = 9p,1I9ot3 - gJ.
akfk~ kj k~ (EJ.
+
Ep,ClI>...,Eyf3l p/J
1f JI1I
+
:>..p(k t , k2 , k3 , kot ) =
E}.laIJld ElIt3I~")
bkf k~ kj k~ ( Ep,QljJ'v,E~~>'" E>".,I>..'p' E~P.' + EIJ011 "1I' Er~>..' E ym >..' p' E:~P.'
+ +
EJ.'QIJ."v,E~i'Ep<SI~'p,E~~'), where a and b are two form factors which depend on all possible Lorentz invariant combinations of the momenta.
Ex. 12.8. 1/128 approximately. Ex. 13.12. Q. = Eab' Ex. 15.13. -17%. Ex. 15.19. +1.9%
J
3
.
d x tPbtP,.
Index IPI diagrams, see irreducible
anticommutator, see anticommutation
diagrams IPR diagrams, see reducible diagrama
relation antilinear operator, 212-214, 219 anti-neutrino, 2, 74
2-point function, 254, 259, 267-269, 271,272,274
antiparticle, 39-41, 45, 67, 71, lOS,
123, 182, 207, 327, 361 antisymmetric symbol, 291, 360 antisymmetric tensor, 49, 65, 354,
Abelian group, 291 accidental symmetries, see symmetries, accidental
364-365 antisymmetry of fermions, lOS, lID,
action, 15-27, 72, 151, 152, 155, 167,
176
168, 285
anti-unitary operator, 213, 215 approximate symmetries, see symmetries, approximate asymptotic series, 80 axial gauge, 151
adt representation, see under group representation amplitude, see Feynman amplitude anapole moment, 229, 230 angular momentum, 26 eigenstates, 65 of a spinar field, 49, 53, 54 orbital, 54 annihilation and creation operators,
3-5, 34-39, 44, 45, 65, 66, 72, 76, 82, 83, 86, 88, 94, 97, 102, 158, 160, 162, 167, 170, 194, 202, 297, 300, 302, 307, 309, 332
axial vector, 206, 207, 218 Baker-Campbell-Hausdorff formula.
291 bare fields, 273, 274 Lagrangian, 273, 274, 277 parameters, 274, 277
baryon number, 167, 289, 35&--357 l1-decay, 2, 74, 32&--328
dimension of, 95 anomalous magnetic moment, see magnetic moment,
constant, 74 bi-unitary transformation, 341 Bianchi identity, 318 bilinear covariants of fermion fields,
anomalous anticommutation relation, 5, 48, 86,
124, 220 of Dirac matrices, 48-50, 54, 59, 120, 209, 215, 239 of fermionic fields, 66-68, 84, 97, 102, 103, 212, 220, see also antisymmetry of fermions
206,211, 216, 217, 219, 220, 226,354 BremBStrahlung, 197-199, 27~282 broken symmetries, see spontaneous symmetry breaking Cabibbo-Kobayashi-Maskawa matrix,
342
and exclusion principle, 68
373
374
Index
canonical commutation relations, 30, 32, 33,38 momentum, 19-21,29,30,32, 38,150,153,161,295 qU8J\tization. 29, 80, 153 center-of-m8.9S frame. see frame,
eM
charge conjuga.tion, 201. 207-212, 214,217-219,250,283,294 charge quantization, 169 chiral representation, see representation of gamma matrices chirality, 335, 336, 338, 340, 341, 352, 357, 360 projection operators, 61-62, 337, 353, 360 color, see qua..rk color Compton scattering, 173, 185--193 Coulomb gauge, l51 Coulomb interaction, to, 163, 194, 200, 204 modification at higher order, 275-276 counterterm, 252, 267-277, 280, 282 symmetry, 217-218, 338 T symmetry, 218-221 T theorem, 220 creation operators, see annihilation and creation operators cr088 section, 41, 115 for 2-to-2 scattering, 133-140 for individual processes, see under the proau name general formula, 130-133 Lorentz invariance of, 144 crossing symmetry, 186, 190 decay rate, 41 general formula, 115-117, 140 of a scalar, 87-94, 97-101, 106, 112,117-122 of Higgs boson, 346 of muon, su muon decay of pion, .lI~e pion decay of W -boson, 345 of Z-boson, 342-345 degenerate vacua, 297, 299
derivative interactions, 88, 130, 246, 307 differential cross section, 137, 139, 142-144, 179,182, 192,193,197-199, 243, 244, 348, 350 decay rate, 129, 130, 140 dilatation, 27 dipole moment, se~ electric dipole moment &. magnetic moment Dirac equation, 1,2,49,51-58,361 covEU"iance 0(, 52-54 plane wave 8OIutions of, 54-58 Dira.c Hamiltonian, 48, 65 Dirac matrices contraction formul8.9, 54, 178 in N-dimensiona, 265 general properties, 48--50, 359 hermiticity properties, 50 non-uniqueness of, 50-51, 359 representations, see representation of gamma matrices trace formulas, 49, 119--120, 361-364 DiTfw:·Pauli representation, .tee representation of gamma matrices discrete symmetries, see under symmetries electric dipole moment, 228-230 electron mass, 144, 339 electron·electron scattering, 199 in QED, 173-179 with sceJ.ar exchange, 101-105 electron-photon scattering, see Compton scattering electron-positron annihilation to muon pairs, 173, 182-1&4, 346-348 to W·pairs, 351-354 electron-positron scattering, 173, 180-182 electron-proton scattering, see und~r proton electrowcak theory, 326-357
Index
375
energy-momentum tensor, see stress-energy tensor equivalence theorem, 353 Euler·Lagrange equations, 12-14, 17-18, 27,63,64, 14B Euler-Mascheroni constant, 264 evolution operator, 75-80, 82, 268 exclusion principle, see Pauli exclusion principle Fadeev-Popov ghosts, 322 Fermi constant, 123, 350 Feynman amplitude, 105- J10, 113,
117, 120, 121, 123-125, 135-137,141,142,144,175, 176, lBO, IB3, IB4, IB6, 195, 19B, 199, 241, 242, 244, 246, 247,254, 2BO, 305, 30B, 310,
343, 349 Feynman diagram, 87-112, 155,172,
175, 197, 230, 309 Feynman parameters, 231-233, 235, 23B, 241, 256, 261, 266, 367 Fcynman propagator as Wick contraction, 85 (or Dirac fields, 69-71 for ghost fields, 323 for massive vector hosans, 150, 309 for non-Abelian gauge bosons, 322, 334 for scalar fields, 41-46 for the photon field, 154
for unphysical scalars, 309, 334 for unstable particles, 355-356 !'eynman rule, 106-110, 114, 170,254 for charged current interactions, 336, 342 for derivative coupling, 130 for effective vertex, 223 for electroweak theory, 339, 353 for external field, 195, 196 for external lines, 107, 164 for ghos"', 323 for Goldstone hosans, 307 for internal lines, 107, 164 for loop integration, 108
for neutral current interactions, 337, 33B, 347, 349 for QED, 170, 173, 175, lBO, 230,
254,261 for vertices, 110
for YM theories, 31B-32O, 322, 352 Feynman-'t Hoeft gauge, 155, t61,
162, 164, 323, 349, 352, 356 fine-structure consta.nt, 10, 179, 233 Fock space, 36, 37, 6B, 77, 94
form factor, 224-244, 246, 251, 259 Lorentz invariance of, 224 4-fermion interaction, 74-75 as low-energy electroweak limit, 350 inelastic scattering with, 140-144 muon decay with, 122-130 (rame eM, 127, 134-137, 142-144, 17B-179, IBl, IB3
Lab, 134, 137-140, 143, 18B, 189, 195, 243 full Lagrangian, 273-275 functional, 17, 19,20,64 derivative, 19,20 Poisson bra.cket of, .'lee Poisson bracket fundamental representation, jee under group representation 'Y-matrices,
.'lee
Dirac matrices
gauge boson, 168, 2B4, 301>-311, 315, 31B, 319, 322-324, 327-330, 332-334, 336, 337, 342, 344, 353 self-interactions, 319-320 gauge covariant derivative, 169, 274, 30B, 314-319, 330, 336, 33B
gauge fixing, 20, 151-154, 176, 267, 274, 275, 309, 310, 322, 334 gauge pri"nciple, 167, 168, see auo gauge b080ns & gauge covariant derivative gauge theory Abelian, jee QED non-Abelian, see non-Abelian gauge theory
Index
376 GaU88 theorem, 17, 18, 22
generation, 335, 340 generators, 289-293, 302, 304, 305, 313,315,317-319,327-331, 336 ghost fields, au Fadeev-Popov ghosts global transformations, su under symmetries gluOlI8, 325 Goldstone bosons, see
internal symmetries, see symmetries, internal intrinsic charge conjugation, 207, 208 ,217 T,22O parity, 202, 203, 205, 206 time reversa.!, 214 ineducible diagr8l1lS, 112 "","pin, 289, 293, 294, 327
Nambu-Goldstone bosons Goldstone theorem, 300-304 Gordon identity, 56, 223, 225, 234 gravity, 324, 326 Green's function, 41, 43, 44, 149, 151, 154, 163, see a130 Feynman propagator group
algebra, 291, see also structure constants general properties, 284 representation, 292-293
adt, 293, 315, 322 fundamental, 292, 313, 314, 316, 317 singlet, 292, 319 Cupta.-Bleulcr formalism, 160-162 hadron, 326, 327 Heavisid~Lorentz
units, 10, J46
helicity eigenstates, 61, 62, 351 of neutrinos, .fee neutrino helicity projection operators, 60-62 Higgs booon, 75, 335, 339, 340, 346 field, 339 interactions, 338 multiplet, 330, 33B neutral, 349 Higgs mechanism, 308-311 hypercharge, 329, 330, 336, 357
'in' and 'out I states, 82 inelastic processes, 115, 140-144, )73, 184, 243 infra-recl. divergence, 198-199, 245, 271, 279-282 interaction picture, 78
Jacobi identity, 293 Klein-Gordon equation, 28, 29, 41, 42, 47,54 with a source, 41 Klein-Nishina formula, 192 Kronecker delta, 6" Lab frame, see frame, Lab Lamb shift, 276 Landau gauge, 155 Lande g-factor, 227, 228 for the electron, 238 for the proton, 242 lepton number, 167,356-357 leptons, 130,324,335,336,340, 341, 346, 356 Levi-Civita tensor, see antisymmetric tensor Lie grou ps, 290 lifetime, 10, 11, 117, 122, 130,355 loop diagrams, 93, 94, 100, lOS, 230, 245, 249, 250, 252, 254, 257, 259, 260, 267, 274, 280, 323 evaluation of, 231-241 Feynman rule, 109, 110 loop expansion, 268 Lorentz force formula, 8 Lorentz invariance. 126, 144, 149, 160, 172, 2OS, 220, 224, 253, 257, 259,277, 317, 326 and VEV, 300 of action, 24, 73 of cross section, see under cross section of Feynman amplitude, 144, 244
377
Index Lorentz transformation, 7, 16, 25-26,
28, 30, 47, 52, 53, 156, 201-203.283.285 indices, 54, 123, 125, 220, 343, 344 of co-ordinates, 7 of fermion bilinears, 54 of spinoTs, 52-54, 73, 208, 209,
211, 257, 287 of vector fields, 156 Lorenz gauge, 151, 185
magnetic moment anomalous. 228-238, 242, 259,
272. 294 Dirac, 227 Majora"a representation, see representation of gamma matrices
Mandelstam variables. 144-145, 179, 183, 346 mass matrix. 341 Maxwell equations, 8, 18, 40, 146,
147, 150. 152. 153, 155. 160. 165, 318 discrete symmetries, 204. 208 minimal substitution, 169, 253 momentum
conjugate, 14, 150 muon, 279, 324, 340 coupling with the Z-b06On, 346
decay, 10, 122-130. 141.350.354 magnetic moment, 238
mass, 10, 144 pair production . .see e- e+ annihilation
Nambu·GoJdstone boson, 303-305 interactions. 305-308 natural units, 9-11. 28, 29,117
neutrino. 123, 126, 127, 129, 130.324. 327,335-337,340,351.356, 357
coupling with the Higgs booon,
340 helicity, 141,335 mass. 124,336 masslessness of, 335, 336, 339
neutral current interactions, 337
scattering off electron elastic, 349-350 inelastic, 140-144 neutron, 2, 74, 75, 288, 289, 293, 294, 324. 326-328. aUo
I.'
P-decay mass, 289, 293 Noether charge, 24, 67 current, 24, 39, 64, 321 theorem, 21-21, 40, 167,200,
223, 283, 30 I non-Abelian
gauge theory, 312-326. 329 group, 291 symmetry,3()()'·301 non.perturbative effects, 357 normal ordered charge, 67 Hamiltonian, 36, 41, 67,161,110 momenLum, 36
product. 89. 91. 92.102.174 and time ordered product, see Wick theorem of bosonic fields, 35 of fermionic fields, 66, 67 normalization of fields. 30, 288 of flux, 132 of hypercharge, 330 of polarization vectors, 185, 344 of spinors, 56, 58, 66, 61, 225,
226 of stales. 5, 35, 37, 44. 76. 94-97, 159, 222 nucleons, 75, 288 number operator, 5, 37, 39, 40 off-shell particle, )) 1, 5U 000 virtual particles on·shell condition, 111, )88,235,280
particle, 111, 112, 172, 222, 230, 255,271,276,344,356 renormaliution, 269, 271, 276
pair annihilation, 173
378
Index
pair creation. 173 p.... ity, 201-208, 214, 217-219,231,
242, 244, 257, 283, 294 intrinsic, see intrinsic parity
violation, 230, 231, 338, 354 Pauli exclusion principle, 68
Pauli matrices. 57, 288, 289. 328, 359 Pauli-Lubansky vector, 65 perturbation theory, 76, 80, 82, 106, 130, 250, 253, 268, 282, 357 phase space, 116 (or 2-body decay, 120-122 rOT identical particles, 117. 179 photon charge conjugation, 208 in electroweak theory, 328, 332, 333 masslessness of, 150, 333 parity t.ransformation, 204, 206 polarization, see under polarization propagator, f49-150, 153-155, 162-164 scattering off electrons, see Compton scattering under time reversal, 214-217 pion, 75, 288, 289 decay, 130 decay constant, 130 m....... ,289 scattering off electrons, 244 Poincare invariance, 16, 72, 285, 301 Poisson bracket, 15, 20, 29 polarization atates circular. 158 elliptical, 158 number of, 159, 161, 162 physical, 192, 193 transverse, 162 polarization Bum (or massive vector bosons, 344 for photons, 164-165, 192, 193, 199 polarization vectors, ~ee abo polarization states for massive vector bosons, 344 longitudinal, 344, 352, see also
equivalence theorem
transverse, 344 for photons, 156-159, 161, 184-189, 192, 193 and gauge invariance, 184-185,
187 longitudinal, 157 scalar, 156. 157, 161 transverse, 157
pole of a propagator 309, 310, 333 principle of least aelion, 12, 13, 17 Proca Lagrangian, ISO, 308 projection operators, 59-62 for chirality, see under chirality for energy. 59-60 for helicity, see under helicity I
for spin, 62--63, 354
propaga.tor. see Feynman propagator proton, 2, 74, 75, 167, 288, 293, 294,
324, 326-328 electric charge as unit, 10. 166, 288 magnetic moment, 242 mass, 289, 293 scattering with electrons, 241-244 pseudoscalar., 205, 206, 212
QeD, 323-325 QED, 166-199, 245, 248, 283, 284, 286, 312, 313, 315, 318, 323, 326, 329, 333, 338, 346, 348 divergent amplitudes, 2:)0-252 invariance under discrete symmetries. 206, 212, 216, 218, 294 renormaJiza.tion, 253-282 vertex function, 230-241 quantiza.tion canonical, see canonical quantization first, 37 second, 37 Quantum chromodynamics, su QeD Quantum electrodynamics, see QED quark color, 324 masses, 279
Index
379
mixing matrix, 342 quarks, 241, 324, 335, 34ll-342, 356, 357
redudble diagrams, 1l2, 271 regularization, 252, 26~267 and gauge invariance, 263
cut-off, 252, 253 dimensional, 252, 26ll-265, 267, 272, 323 Pauli- Villars, 252, 265-267, 272 relativity general theory, 326 special theory, 5-8, 301 renormalizability non-renormalizable theories, 248, 275, 326 of electroweak theory, 334 renormalizable theories, 248, 253 representation of gamma matrices chirsl, 360
Dirac-Pauli, 60, 203, 210, 226, 228, 229, 359--360 spinoT solutions in, 56-58, 226 Majorana, 210, 361 running coupling constant, 27&-279 Rutherford scattering formula, 197,
275
R( gauge, 334 S-matrix, 8ll-86, liS, 116, 130, 131, 155, 172, 174, 175, 180, 184, 185, 194, 230, 275 scalar electrodynamics, 169 scattering angle, 135, 142, 143, 181, 354 self-energy for fermions, 174.251,254-157, 265-267, 27ll-272 for non-Abelian gauge fields, 323, 324 for photon,
see. vacuum
polarization simple harmonic oscillator, 3-5, 32 36 singlet representation, see under group representat.ion spin of a Dirac particle, 65
spin sum, 56,118-120, 176,177,181, 343, 353, 354 spont.aneous symmetry breaking, 294-311,330,331,333,338, 339 standard electroweak theory, see clectroweak theory step function, 31 stress· energy tensor, 23, 25, 21 structure constants, 291, 292, 314, 317,320,323,329 SU(2) definition, 287 doublet, 327, 328 generators, 290-292 isospin, 289 n-plct, 330 transformation, 289, 290 SU(3), 324 325 superficial degree of divergence, 245-251 symmetries accidental, 351 and conservation laws, su Noether theorem approximate, 289, 293-294 continuous, 200, 283, 284, 286-291 breaking of, 299-301, se~ also Goldstone theorem, Nambu-Goldstone bosons discrete, 20ll-221, 283, 294, 295, 298, 304 global, 148, 166-169, 284, 286-289, 299, 308, 310, 312-315,324,331, :156-357, see also Noether theorem internal, 16,26··27,283,285-281, 301 srace-timp., 24-26, 283 tau, 279, :l40 temporal gauge, 151 Thomson cross·section, 193 t.hreshold energy, 144 time dilaLiOll, 122, 355 time reversal, 'lOI, 212-217, 219,283, 294
380
Index
time-ordered product, 79, BO, 174
and normal-ordered product, see Wick theorem for fermion fields, 11
for scal", fields, 45 total divergence, 18, 23, 24, 64. 73, 148, 15], 309, 334 tree level, 100,253-255, 267, 269, 271, 274, 277, 305, 306, 323, 327, 337, 351 interaction, 225, 241
ultra-violet divergences, 241, 245-282 uncertainty rela.tions, 9, Ill, 355 unitary gauge, 310, 311, 333, 334
We;nberg angle, 333, 337, 345 Wick contractions, 84 expansion, 86, 87, 90, 91, 94, 98,
104, 174 theorem, 82-86, 88, 101, 106, 174 Wick rotation, 235-237, 367 Yang-Mills theories, ~ee non-Abelian gauge theory
YukaW8. coupling, 339-34l interaction, 75,87,101,117,256, 338 theory, 75, 104, 288
unitary groups, 327 unitary matrices, 50, 51, 210, 215,
286-288,290,313,341, 359
as a group, 285, 290 unitary operator, 76, 81, 213. 215
units,
8U
natural units. Heaviside-Lorentz units
unphys;cal mode, 303, 310, 322, 333, 334, 339 vacuum expectation value, 35, 84, 85,
98, 297, 299, 301, 304, 332, 339 of normal ordered products, 82 of time-ordered product, 84
vacuum polarization, 174,251, 257-265, 267, 270 vacuum stat.e. 34, 36, 37, 40, 41, 45, 68, 82, 83, 89, 94, 95, 97, 102, 103, 107, 159, 297, 299, 301,302,331,332 vacuum to vacuum transition, 94, 174 vertex function, 223-224, 230, 242, 244, 253, 255, 258, 259, 272, 280 virtual particles, 1U}-113, 172, 282. 322, 323
VV-boson, 279, 327-354 decay of. sa decay of W -boson mass, 332, 345 VV",d-Takahashi identity, 253-256, 258, 272, 274, 276
Z-boson, 207, 2l7, 332-353, 355 decay of. see decay of Z-boson
decay width, 356 mass, 345