Properties and Production Spectra of Elementary Particles

176 Pages·1972·13.446 MB·English

by  H. Pilkuhn (auth.), H. Schopper (eds.)

Ref. p. S] 1.1 Particle properties: Introduction 1
I Particle properties, coupling constants and form factors
1.1 Introduction: The basic interactions
1.1.1 Units and conventions
The following units are normally used in particle physics
h = c = 1 = 6.582183. 1O-22M eV set = 197.3289M eV fm, (1)
where 1 MeV is lo6 eV or 10e3 GeV (or BeV) and 1 fm (fermi) is lo-l3 cm. The fine structure constant CIz 137-l
is written as e2/4n.
For cross sections one uses
GeV2 = 0.38935m b, 1 mb (millibarn) = 10m2’ cm2, lt.tb=10-3mb. (2)
Other fundamental constants are collected in 1.4.1. The conventions about metric, state normalization, Dirac
matrices etc., are not uniform. We shall use the 4-momentum vector P, energy E, particle massm , and metric tensor
glly as follows:
P =(P")=(P,,~)=(E,p,,p,,p,), P2 = Ez-p2 =m2,
i1 0 1
-1 (3)
gl,v = P;P, =p,"p;gsy =b;Pbp=Pa(lP&
-1 ’
\o -I/
We define the connection betweent he S- and T-matrices and the normalization of i-particle statesi n a Lorentz-
invariant way:
Si, = (f/i) +i(27~)~s 4(Pi-Pf)Tif, (4)
(~~l~i)=(2K)32ErS~~f-i).
The Dirac equation, y-matrices and spinor normalizations are taken as
P.yu=mu, 5=14+y", i@4')u(M)=6,,.*2m,
where A4 denotest he magnetic quantum number. The relation betweent he Dirac spinor u and the Pauli spinor x is
u=(E+m)+ (7)
For an outgoing antiparticle of 4-momentum P and helicity A(= spin component along $), we simply use the nega-
tive-energy spinor u( -P, A). For photons and vector mesons,w e need the spin-l states s(A):
P.&(l))= 0, &(A).& *(A)= - 1 ) (8)
s(f1)=2-f(O,rl,-i,O), s(O)=+(P,O,O,E). (9)
Textbooks using the samem etric are: [Schweber 61; Muirhead 65; Bjijrken 65,67; Gasiorowicz 66; Pilkuhn 671.
Other metrics are used by [Martin 70; Klllen 64,65; Marshak 691. Some books take -i instead of +i in (4), some
omit the factor 2E, in (4) and some the factor 2m in (5). The latter factor is used to make the momentum space
(seee .g. Eq. (4) in 1.1.5b elow) the samef or fermions and bosons. Some books take all y-matrices hermitean. The ys
in (6) is sometimes called iy,, or sometimes even - ys (for example in [Marshak 691).
For spins 3 and 2, the Rarita-Schwinger spinor u,(A4) and tensor .sa#4) will be used:
U,(+~~=U(+;bp(+_l), U,(+-~)=3-+U(T~)&ll(+-1)+(~)QU(+_~)&fi(0). (10)
E,,v(+-2)= E,,(+_1 ) E,(+ 11, E&1)= 2-f[&p(+--)EY(0)+E~(O)Ey(+_1)],
(11)
~~~(0=) 6-“[s,(l) sY(- 1) + EJ- 1) e,(l) +'240) EJO).]
Pilkuhn
I Landolt-Bihstein, Neue Serie l/6
2 1.1 Teilcheneigenschaften: Einleitung [Lit. S. 8
The resulting propagators arc collected in 1.3.3.F inally, we shall frequently need the triangular function ). for the
calculation of momenta in the a. b, or c rest frames. For A &-& , + p’,= 0,
pqpi = 0) = (41f13- l i,(& n& IfI:), i = a, b, c:
(12)
i(o,b,r)=~2+b2+~2-2(ab+bc+ca)=[a-(~+1/E)2][a-~-~)2].
Constants which arc closely related to experiment are normally adorned with a statistical error. This error is the
l-standard deviationu ncertainty in the last digits ofthe constant and will be given in parenthesesim mediately after the
digits. for example 0.72(S)= 0.72f 0.05, 0.72(11)= 0.72 f 0.11, 7.2(1.1). 10e4 = (7.2f 1.1). 10m4.W here experi-
mental information is lacking (e.g. X0 lifetime) or obviously less accurate than the theoretical prediction (e.g.
branching ratio heE/A-f in X0 decay, or some branching ratios of resonance decays that follow from isospin-
invariance + phase space).t hcorctical values are given, followed by the abbreviation “(th)” to distinguish them
from experimental values. Whenever possible. I have adopted the recommcndcd values of the “Compilation of
coupling constantsa nd low-energy parameters” [Ebcl71], the “Reviewo f Particle Properties” [Particle 71,721 and
other review articles. In these review articles the reader will find further referencest o the original literature. Also,
in this very compact tabulation it is not possible to enumeratea ll the additional effectsw hich may occur for certain
reactions or decays.F or this reason, I have quoted relatively many reviews. I have used the references ystem of
[Ebel 711 and [Particle 711, which gives the name of the first author and the year of publication as referencek ey,
and the journal abbreviation. volume and page numbers,a nd list of authors in the list of referencesT. he journals are
abbreviated as follows:
AP Annals of Physics
FP Fortschritte der Physik
JETP English translation of Soviet Physics JETP
JETL JETP Letters
NC Nuovo Cimento
NCL Nuovo Cimento Letters
NP Nuclear Physics
PL Physics Letters
PPSL Proceedingso f the Physical Society of London
PR Physical Review
PRep Physics Reports
PRL Physical Review Letters
PRSL Proceedingso f the Royal Society of London
PTP Progresso f Theoretical Physics
RMP Reviews of Modem Physics
SJNP Soviet Journal of Nuclear Physics
STMP Springer Tracts in Modern Physics
ZP Zeitschrift fiir Physik.
Finally. I wish to acknowledget he help of my colleaguesa t Karlsruhe, particularly of D. Wegeneri n connection
with the electromagnetic for factors.
1.1.2 The classification of particles and resonances.S tate mixing
Particles may be classifiede xperimentally according to their lifetimes.T ab. a showst he classeso f stable particles,
weakly decaying particles (lifetimes >0.8 . 10-l’ set) and electromagnetically decaying particles (lifetimes
10-l” . . . lo-l9 XC). Our knowledge of these three groups has reacheLda certain completenessi n recent years.
There is a forth group of extremely shortlived states, normally called “resonances”.H ere only the lower-lying
states are known with some accuracy, and a complete tabulation is premature. Among these resonances,I have
included the nonets of vector and tensor mesonsa nd the baryon decuplet, which form complete SU(3)-multiplets,
and four additional baryon multiplets which are nearly complete. From the remaining resonancesI have included
only a few low-lying states with not too broad widths. An exception is the c-meson,w hich is presently not even
confirmed as a resonance,b ut which is used in the description of nN and NN scattering. Complete tabulation of
all resonancesi s given in the “review of particle properties”, which is published at regular intervals [Particle 713.
On the other hand, I have included some “elementary particle” properties of the nuclei d, t, 3He and a in the
appropriate places.
Pilkuhn
Ref. p. S] 1.1 Particle properties: Introduction 3
Tab. a). Classification of particles according to their lifetimes
Stable particles Weakly decaying particles ‘)
Name Symbol Spin Antiparticle Name Symbol Spin Antiparticle
photon Y 1 -1 muon p- or p f p+ 0rjI
electron e-ore 3 e+ arc neutron 4 ii
electron-neutrino v, 4 ve lambda-hyperon : i il
muon-neutrino ‘) V~ t VII sigma-hyperons z+,z- 3 z-, e+
zo -- 1 EO z+
proton P 4 F Xi or cascadep article , a T ,-
deuteron d 1 a omega-particle n- t ai
1
helion or helium-3 3He 3iT2 3) charged pi-meson or pion rc+ 0 K-
a-particle (helium) 4He or a i 4Ez 4) K-mesons or kaons K+,K’ 0 K-J70
Electromagnetically decaying particles:
z” and E”, rc” and rl (q-meson). Limiting cases:X (X-meson or r-l’s, ee 1.4.2)a nd o (omega-meson,s ee 1.4.3).
‘) The distinction betweenv , and vp is of importance only in neutrino reactions: v, produces electrons,v ,, produces muons.
2, The lightest weakly decaying nuclei are the triton (t or aH, see table in 1.4.6~)a nd the hypertriton 2H. The latter has
a binding energy of only 100k eV and a lifetime of 2 - 3. 10-r” set (see [Ram 713 for example).
3, The first observation of this particle is published in SJNP 12, 171 (1971).
4, Not yet observed.
A more theoretical classification of particles and resonancesi s provided by their quantum numbers. The
lepton group consists of e, p, v,, v, and their antiparticles; the photon is in a group by itself; the remaining states
have strong interactions and are called hadrons. The hadrons are subdivided into baryons (which are fermions) and
mesons (which are bosons). They occur in multiplets of 2Z+ 1 states (Z = isospin), the members of which are
distinguished by their charge Q, which is related to the third component of isospin Z3a s follows:
Q=Z,++Y (Y=S+B). (1)
The hypercharge Y (= twice the averagec harge of the isospin multiplet) is a new quantum number which is con-
servedb oth in strong and electromagnetici nteractions. It is equivalent to the older quantum number S (strangeness),
which is defined such that nucleons and pions have S = 0 (B = baryon number: + 1 for baryons, - 1 for antibaryons
and 0 for mesons).H adrons with Y + 0 will be called “hypercharged”,t he others “hyperneutral” (the corresponding
names with respect to S are “strange” and “nonstrange”).H adrons also form approximate SU(3)-multiplets of
multiplicity 1 (singlet: I = 0), 8 (octet: Z = 0, 2 x (I = $), Z = l), and 10 (decuplet: Z = 0, ?, 1, i). This classification
is shown in Fig. 1. For each singlet, there exists an octet of identical spin and parity and similar massesT. he SU(3)-
breaking part of the strong interaction induces a certain mixing betweent he two Z = 0 states,i .e. the physical states
rl and X =rl’ of the pseudoscalarm esons‘ ), $ and o of the vector mesons,f and f’ of the tensor mesonsb ecome
linear combinations of the octet and singlet states( denoted by 8 and 1):
q = qs cod, - X, sine,, ~=~scos~,-o, sine,, f=fscos&-fisinf?,,
(2)
X=rlssinO,+X,cose,, 0 = I$~s ine, + w1 case,, f = fs sine, + fi case,
with approximate mixing angles of lo”, 40” and 32” (seet ables in 1.4.2 . 1.4.4).T hesea nglesa re obtained from the
Gell-Mann-Okubo mass formulas. For the pseudoscalar mesons,
4m,Z = m,Z+ 3(m,’ c02e, + rng sir?@,). (3)
For baryons, linear mass formulas are used, e.g. 2m, + 2m, = m, + 3m,. We shall call the mixed singlet-octet a
“nonet”, although this term is frequently reserved for meson nonets in the quark model’) [Kokkedee 691. The
experimental determination of particle quantum numbers is explained for example in the books of [KHllen 64,651,
[Cool and Marshak 681,o r [Miller 691. Specialb ooks on SU(3)-symmetry are [Gourdin 673 and [Carruthers 661.
A treatment in terms of U-spin (U-spin multiplets have fixed Q instead of fixed I) is given by [Pilkuhn 67J.
‘) At the time of writing, the spin of X could also be 2, in which case Qp= 0.
‘) The signs in Eq. (2) are chosen such that sin6 = +3-i in the quark model. Seea lso 1.3.4d.
Pilkuhn
I*
4 1.1 TeilcheneigenschaftenE: inleitung [Lit. S. 8
I
baryan decuplet baryon octet pseudoscolor octet
0p .K” .K+
1 0 lA 0 ;+ ;++ l n
A-
t
:;I, , :i’:.:“, , , : , :I’ , I .IK - I .iI? ’ I
-3/l -1 -‘I2 0 l/l 1 3/2 -1 42 0 ‘/2 1 -1 -‘/2 0 ‘h 1
13- 4-t 13-
Fig. 1. (Y, I&diagramsf or the baryond ccupletb, aryono cteta nd pseudoscalamr esono ctet.T he octets tateq s is definedin
Eq.( 2)i n 1.1.2T. heo ctetso f the vector( K*, e, $8, i(*) andt ensor( KN,A ,, f,, R,) mesonsa rea nalogoutso the pseudoscalar
mesono ctet.T he symbolI * representtsh e x*(1385)r esonancea.n d 8* standsf or the 2*(1530)r esonanceT.h e Y = 0 row
containst he hyperneutrapl articles.
The neutral kaon statesK ” and if0 have opposite hyperchargeb ut otherwisei dentical quantum numbers.A s
hyperchargeis not conservedi n weak interactions, the statesh aving definite lifetimes in a sufficiently thin medium
(K, = shortlived kaon, K, = longlived kaon) are linear combinations of K” and R” of the type
K,=2-f[(l+~)K+(l-E)R], K,=2-*[(l-c)K-(l+~)R-j, (4)
with I&l+ 1 (seeT ab. 1 in 1.4.2,n otesfand g). This mixing is physically different from (2) becauseit takes a time
interval of > lo-” set to convert a state with definite hyperchargei nto a state with defmite lifetime. Otherwise,
statem ixing by weak interactionsi s negligiblee xceptf or the theory of weak decays.S tatem ixing by electromagnetic
interactions occurs betweenA and Z” and between1 7a nd no:
A = A1 COST,,+ Z, sin6,,, q = qI cosO,, + n3 sin&, ,
(5)
Co = -A, sinfI,,+ C, cos0,,, 7r” = -qI sin&,+n, cos&,
wheret he subscripts1 and 3 denotet he pure SU(3) singlet and triplet states.A s thesem ixing anglesa re small, one
can replaces in0 by Band cosfI by 1.T hen the SU(3) formulas for electromagneticm asss plittinggive(see [Dafitz64];
[Carruthers 663 or [Gourdin 677)
2
mp-m,+m,0--m,+ is
em= l/j(,,fA-m3 , enq= mK* -m zfmf-mi+ . (6)
I/rCq - 4)
Particlest hat carry no chargew hatsoevera re eigenstateso f the chargec onjugation operator C. The neutral vector
mesonse ”, o and Q have C = - 1 like the photon, and no, 9, X, A$!,f and f have C = 1. As a consequenceo f charge
conjugation and isospin invariance, all hyperneutral mesonsa re eigenstateso f the G-parity operator
G = C. exp(inl,) (7)
with eigenvalue+ 1 for e*, e”, q, X, f and f and - 1 for n*, x0, o, 4, A: and A;.
1.1.3 Electromagnetic interactions
The Hamiltonian of the electromagnetici nteraction is
NC,,,= eA&,,, j:,,, = qe)VY,,) + quJVY’,++, uj:,,,( hadr) (1)
where A,, is the 4-potential and j:,,, the electromagneticc urrent operator. The electron and muon componentso f
jz,,,a re usedi n quantum electrodynamics.T he hadronic partj$Jhadr) is not known, but therei s somee videncet hat
it hast he isospin- and SU(3)-propertieso f the electric chargeo perator Q, i.e. it is a mixture of an isovector operator
(I,) and an isoscalaro perator (Y). One may thus say that the photon behavesa s a mixture of an isovector and an
isoscalarv ector meson.T he vector-meson-dominancem odel (VMD) explains this by a dominanceo f the low-lying
vector meson states (e,(x) = e-mesonf ield etc.):
(2)
A’,(hadr)
Pilkubn
Ref. p. S] 1.1 Particle properties: Introduction
SU(3)-symmetryp redicts the photon to behavel ike the U-spin singlet member of an octet (i.e. like 6~ + Cps)I.n
practice, field theory is only neededf or radiative corrections. We shall be concernedd irectly with the T-matrix
element (Eq. (4) in 1.1.1).F or the emission of a photon, 7’ is of the form
Tcf) = es:(A) Tfl with P,,, Tfl = 0, (3)
with P, = 4-momentum of the photon and E: the complex conjugateo f Eq. (9) in 1.1.1.T he seconde quation in (3)
may be called gaugei nvariance; it is related to the conservation of the electromagneticc urrent. For the decay
Co+ Ay, gaugei nvariance and parity conservationr estrict T” to the form
(4)
with t = Pyz= 0 for a real photon and F,(t) = 1. Similarly, the matrix elemento f the electromagneticc urrent between
identical baryons b of 4-momentum P in the initial state and 4-momentum P’ in the final state is given by
=ub[(F,+~F,)y,-IcF,(2m)-'(P+P')~ub,
t = (P -P’)‘, Fl (0) = Q, F2 (0) = 1, e(Q+ rc)/2m = p .
where p is the magneticm oment. One frequently writes t = -9, where 4 is the momentum transfer in the ems,i n
the casem b = mbT.
In general,f or processesh aving only one hadron in the initial and final state,T , is determinedb y a few“ coupling
constants” (eQa nd eK in our example)f or a real photon. When the photon is taken off the masss hell,t hesec oupling
constantsa re modified by “form factors” Fi(t).
It should be noted that (5) is the relativistically correct form for electron scattering on any target of spin $. In
particular, the formula also appliest o 3He, with m = m (3He)i n (5), and in Eq. (7) in 1.1.1.O n the other hand, the
interpretation of a form factor as the Fourier transform of a spatial charged istribution is a,nonrelativistic concept
which becomesu selessf or large 4’.
For pions, we have
TJ4 = Q,@,(t)( p, + K),, , F,(O)= 1. (6)
In particular, the rt” meson( anda lso q and X) cannot emit or absorbp hotons,d ue to chargec onjugationi nvariance,
except by transforming themselvesin to e”, o or 4 mesons( see1 .4.3~).
The completem atrix elemento f electron-hadron-scatteringis (seeE q. (1) in 1.3.3f or m, = 0, and Eq. (3) in 1.3.3
with gaugei nvariance (3))
T(eb-+e’b’=) e%~y%,TJt . (7)
In addition to interactions obviously involving photons, there are decayss uch as ce--tn+n - or n +xzn which
do not conservei sospin but do conserveh yperchargea nd parity. Their matrix elementsa re roughly a factor
CI= e2/4n= l/137 smaller than those of strong interactions; for this reason they are also called electromagnetic.
They could be induced by emissiona nd reabsorption of a virtual photon. In this caset he isospin could changeb y
as much as 2 units, but it appearst hat the A I = 2 piece is suppressedT. his is analogoust o the nonleptonic weak
interactions (seet he following section). For a recent review of the electromagnetici nteractions of hadrons, see
[Cumming 71-J.
1.1.4 Weak and superweak interactions
The Hamiltonian of the weak interaction is probably
3 weak = 2-*gj$p+, j, = ~wY,(~ - ~4 yv:, + ~c,,~,(l - YA 'Y,, +j,Wdr) , (1)
whereg is the weak interaction constant measuredi n muon decay.T he word “probably” must be addedf or three
reasons: Firstly, neutrino-electron scattering as predicted by (1) remains to be detected.S econdly,n onleptonic
decayss ucha sA +pn- cannot be shownt o be of the form jJ“ (seeb elow).F inally (1)d oesn ot allow the calculation
of higher order corrections.T herefore,a ll the physics is already contained in the form of the matrix elementf or
emission of an e, V, or p, VPp air:
W,) = W’,4) Y,$ - ~5)u ,,(- PC&> 2;W) ’+ -4 (2)
where Vfl is a vector like T”, and AV is an axial vector.
Pilkuhn
6 1.1 Teilcheneigenschaften:E inleitung [Lit. S. 8
For bnryon decaysb +dlv, V@a nd A’arc given by
(3)
A,,= 2-+7id[-%&) I’,+, BP(~(p) - p’),+, g,(t) (p + p’),lY,&, (4)
where (3) is simply the non-gauge-invariant generalization of TV. Actually, it appears that at least for hypercharge
conserving decays( “hyperncutral decays”),V , is in fact also gauge invariant. This is called the conservedv ector
current hypothesis (CVC). Also, the isospin properties of the charged vector current are identical with those of the
Q* and K*’ states.I t thus appearst hat the vector mesonsd ominate both the electromagnetica nd the weak vector
currents.
The relative strength of hypcrneutral and hypcrcharged couplings is !ixed by Cabibbo’su niversality hypothesis
(0,. = Cabibbo angle):
gy=O = g coso,., g\PyI=’ = g sin& (5)
apart from SU(3)-cocfflcicnts (see 1.4.6h ). Similarly, the axial currents form the charged components of an octet.
Consequently, the following selection rules hold both for V,, and for A,,:
Al=1 for AY=O, Al=f and AY=AQ for lAYI=l. (6)
The relative strength of the axial IA Y( = 1 and A Y = 0 couplings is tan0, z tan0,, but gA +g,, (see 1.4.6a).
The matrix elementsA ,, arc not gauge invariant, but (P - P’),,A” (the matrix elementso f the divergenceo f the
axial current) arc dominated by the pseudoscalars tatesr c* and K *. This leads to the Goldbcrger-Trciman relation
(7)
-&.fbh’n = (8’d-f%~,.bb*
betweent he pion decayc onstant g, (see1 .4.2d ), the axial decay constant gA, and the strong (pseudovector)c oupling
constant fbb.n+ (see 1.3.3a nd 1.3.4).T he corresponding relation for kaons is much more uncertain. The full axial
hgpcmcutral current has negative G-parity like the pion. Hypcrneutral currents with positive G-parity in the axial
part and negative G-parity in the vector part, socalled “second-class”c urrents [Weinberg 583 appear to be absent
except for secondary effects in nuclei [Wilkinson 71; Delormc 713.
The matrix elements of an octet operator, taken between two octet states such as the baryons, contain two
reduced matrix elementsg F and g,, in addition to SU(3)-coeflicients. We shall use
gf -t gD = g coso, BDC8+f BDI- ’ = %c,k. (8)
At r =O. however. the vector current couples to the total charge and has therefore a(weak, vector, f = 0) = 0. Also,
it is so far not possible to measurea ny t-dependenceo f c(.T herefore, a will denote the averagef raction of D-type
coupling of the axial current. The prccisc factors are given in 1.4.6h .
Nonleptonic weak decays such as A-tnN, Z--+nN (1.4.6~) or K-+nrt, K+rrnrc (1.4.2g and h) follow the
Al = ) rule (the decay K * -+K* no being the most notable violation). If these decaysa lso originate from the inter-
action j,jp+ with Al = 1 for the hyperneutral and Al = i for the hypercharged current, then Al = $ should also
result. The suppressiono f the Al = 2 piecei s not understood, similar to the Al = 2 suppressioni n the corresponding
electromagnetic decays.
Interactions which do not conserveC P (P = parity, C = charge conjugation) are so far restricted to the neutral
kaon decays.E ven there, the whole effect seemst o be due to the fact that the states K, and K, in Eq. (4) in 1.1.2
are not CP-eigenstates( they are so for E= 0). If this is true, one can divide the weak interactions into the normal
weak interactions which conserve CP, and into a “superweak”,C P-violating interaction which gives measurable
effects only in K” - iTo-transitions [Wolfcnstein 641.
For the weak interactions in particle physics, the book of [Marshak, Riazuddin, and Ryan 691 can be recom-
mended. (There is at present no comparable book on the electromagnetic interactions of hadrons.) Books on
nuclear p-decay are [Konopinski 663, [Schopper 663 and [Wu 663. Reviews on leptonic decays of hadrons are
given by [Willis 683 and [Rubbia 691. Set also [Bailin 711.
In nuclei, two other forms of weak decaysc an be observedw hich will not be discussedh ere: stimulated A decay
hp+pn in hyperfragments[ Davies 673a nd parity-violating, hyperchargec onservingd ecayss uch as ‘60-+‘2C + 0:
[HIttig 701 or the parity impurities in np-td-( reaction [Lobashov 701. Seea lso PRL 29,518 (1972).
1.1.5 Strong interactions and resonances
The electromagnetica nd weak interactions of hadrons are essentiallyc haracterizedb y a few coupling constants
and form factors. For the strong interactions, the situation is more complicated. Supposef or example that in a
reaction ab-+cd, the exchangeo f a stable meson e of mass 111i,s possible. Taking all particles as spinless,t here
Pilkuhn
Ref. p. S] 1.1 Particle properties: Introduction 7
exists a unique decomposition of the scattering amplitude,
T(ab+cd) = gacegbde(m-zt) -’ + T (1)
where T’ is regular.at t = rnz. The g’sa re the coupling constants. The decomposition is accomplished by means of
dispersion relations (1.35). We can also decompose
T(ab+cd)= V,,, V,,,(mz -t)-’ + T”, v = g . F(t), F(t)=l+F’.(t-m,Z)+ ... (2)
where V are the vertex functions and F the corresponding form factors. Since terms proportional to F’ have no
pole at t = rnt, they are contained in T’ and can be obtained only if a reliable model for T” exists. The product
Va ce. Vbdeis called a residue function. In the Reggep ole model, one tries to approximate T by a sufficiently large
number of “Reggep oles” (see1 .5.6d).T his is always possible, but in that caset he residue functions will not neces-
sarily factorize. If one insists on factorization, no strong reaction is known with T” = 0. This should be a warning
to anyone who tries to use the coupling constants with or without form factors in a single-particle exchangem odel.
Strong form factors will be given only for rt exchange( 1.5.5)a nd for the A resonance( 1.4.7a).A s the stable particle
poles lie outside the physical region, other parameters are of equal or even greater importance in describing the
physical amplitude. In particular, scattering lengths are included in the tabulation.
The situation is slightly better for the decay coupling constants of resonances.C onsider first the weak decay
of an unstable particle. The total decay rate r is obtained from the exponential decreaseo f intensity I in proper
time z = tl,bEl,b/m
dIJdr = -rI, r=t,‘ln2=0.69315t;’
where t, is the half life (r-l is the mean life or just lifetime). The partial rate T(d -+ 1 . . . n) for the decay of particle
d into n particles of spins Sr . S, is related to the decay matrix element by the “golden rule”
r(d+l . . . n)=$JdLips(m’;P, . . . P,) 1 lT(d+1 . . . n)l’,
1,. . .A”
(4)
dLips(m’;ir . ..I’.)= fi -da3(,p- i ~~Ei)S(~~ai)(2S)4-3”,
i=l 2Ei
where m is the mass of particle d, and dLips is the Lorentz invariant phase space element [Pilkuhn 67l in the
ems @d= 0). The summation in front of lT[’ extends over the helicities 1, . . . 1, of the produced particles. Already
for electromagneticp article decays( noI rl”, X0), (3) cannot be measured.F or strongly decayingr esonancesr, can be
measured from the invariant mass distribution of its decay products, which is given by
P(s1 , ,,J = const . mT(d+l . . n) [(m’ - s1, ,,” )’ + m2r2]-1, s1, ,,” = il ‘i ’ 7 (5)
( 1
provided the production matrix element for the resonance (in a reaction ab-+cd say) does not vary between
s1 . . .n -- mz and mz f mr. If r is not too large, r is the full width at half maximum of the distribution (5), and it
does not matter whether one puts m or fi in (5). If on the other hand r is large, (5) cannot be fitted with an
s-independent r. In this case,t he normally adopted prescription is to replace m by fi everywhere in (4) except
possibly in front of the integral. For a decayi nto two particles, the main changei s in the “threshold factor”, r GZp zl+ I,
where p is the actual decay momentum at invariant mass 1/s12i nstead of the decay momentum at the resonance
position md( see1 .2.1).T he resulting P(s1 , ,,“)i s called a Breit-Wigner resonance.A s to the factor .1/mi n front of the
integral, it appearst o be most appropriate not to changei t. If one writes (4) in the form mr = J . . , one has exactly
the combination mT entering (5). (One may of course also write 1/S1-nr in (4) and in (5), which is merely a redelini-
tion of the symbol r.)
If r is large, (5) need no longer describet he invariant mass distribution. In this case,t he simplest starting point
is the decomposition of the T-matrix for the over-all process ab + 1 . . . n,
T(ab+cl . . . n)= T(ab+cd)(m’-s, ,.,” -imI’,)-’ T(d+1 . ..n)+T”. (6)
where the contribution Ti of T” to the resonating partial wave d is negligible around s, ,,,” = m2 - imr. One may
then still determine the whole function r(s I J. If on the other hand no decomposition (6) with negligible Tl exists,
one should read (6) analogous to (l), with T(ab+cd) . T(d+ 1 . . . n) given by their values at m2- imr (residues)
and r given by r(m) according to (4).
It is possible that in such casest he resonancea pproximation is not too useful altogether. It may be difficult to
prove that the residue factorizes. For resonanceso ther than s-wave resonances,T (d+l . . . n) is complex at the
resonancep osition m2 - imr since the decay momentum is complex there.
In practice, ITI is frequently approximated by a sum of distributions of the type (5) in the various invariant
masses( for three particles this is done in the Dalitz plot) plus a constant, noninterfering “background”. More
Pilkuhn
8 1.1 Teilcheneigenschaften:E inleitung
elaborate prescriptions arc used for example in the Veneziano model. Sometimes (5) is used with constant r
although a Breit-Wigner width gives a better lit. In this casen tT(n~)i s somewhat smaller than the “experimental”
fnr. The coupling constants always refer to nlT(nl). Sometimest he resonancef orm r[(a~ -fi)’ + r2/4]-’ is used.
This agreesw ith (5) for fi + tn x 21~a1n d is generally harmless for baryon resonancese, xcept for the A resonance.
The vertices of strong interactions can also be translated to interaction Lagrangians (see1 .3.3).
1.1.6 Textbooks and review articles
Bailin. D. 71 in Reports on progress in physics,V ol. 34.
Bjorken. J. D., and S. D. Drell 65 Relativistic quantum fields (New York: McGraw-Hill).
Bjorken. J. D., and S. D. Drell 67 (Deutsche Ubersetzung) Relativistische Quantenfeldtheorie, B. I.-Hoch-
schultaschenbiicher lOl/lOl a (Mannheim: Bibliogr. Inst.).
Carruthers, P. 66 Introduction to unitary symmetry (New York: Wiley).
Cool. R. L., and R. E. Marshak 68 Advancesi n particle physics,V ol. 2 (New York: Wiley Interscience).
Cumming. J., and H. Osbom 71 Hadronic Interactions of electrons and photons (London: Academic
Press).
Ebel 71 NP, B 33,317: G. Ebel, A. Miillensiefen, H. Pilkuhn, F. Steiner, D. Wege-
ner, M. Gourdin, C. Michael, J. L. Petersen,M . Roos, B. R. Martin,
G. Oades,J . J. de Swart.
Gasiorowicz, S. 66 Elementary particle physics (New York: Wiley).
Gourdin, M. 67 Unitary symmetry (Amsterdam: North-Holland).
KHllen. G. 64 Elementary particle physics (Reading: Addison-Wesley).
KBllen, G. 65 (Deutsche ubersetzung) Elementarteilchenphysik, B. I.-Hochschul-
taschenbiicher 100/100a/100b( Mannheim: Bibliogr. Inst.).
Konopinski, E. J. 66 The theory of beta radioactivity (Oxford: ‘ClarendonP ress).
Kokkedee, J. J. J. 69 The quark model (New York, Amsterdam: Benjamin).
Marshak. R. E., Riazuddin, 69 Theory of weak interactions in particle physics (New York: Wiley
C. P. Ryan Interscience).
Martin, A. D., and 70 Elementary particle theory (Amsterdam: North-Holland).
T. D. Spearman
Miller. D. H. 69 in High energy physics( Academic Press,e d. Burhop, E. H. S.),V ol. 2.
Muirhead. H. 65 The theory of elementary particles (Pergamon Press).
Particle 71,72 RMP 43, S 1, and PL 39 B, 1: Particle data group.
Pilkuhn. H. 67 The interactions of hadrons (Amsterdam: North-Holland).
Rubbia. C. 69 in High energy physics (Academic Press,e d. Burhop, E. H. S.),V ol. 3.
Schopper,H . F. 66 Weak interactions and nuclear beta decay (Amsterdam: North-Holland).
Schweber,S . S. 61 An introduction to relativistic quantum field theory (Evanston: Row,
Peterson& Co.).
Willis, W., and J. Thompson 68 in Advances in particle physics (New York: Wiley Interscience, ed.
Marshak, R. E., R. L. Cool), Vol. 1.
Wu. C. S., and 66 Beta decay (New York: Wiley).
S. A. Moszkowski
Further references for I. I
For journal abbreviation seep . 2
Dalitz 64 PL 10,153: R. H. Dalitz, F. von Hippel.
Davies 67 High En. Phys.( Academic Press,e d. Burhop). Vol. 2, 365: D. H. Davies, J. Sacton.
Delormc 71 NP B 34,317: J. Delorme, M. Rho.
Hiittig 70 PRL 25,941: H. H?ittig, K. Hiinchen, H. WIffler,
Lobashov 70 JETL 11,76: V. M. Lobashov, A. E. Egorov, D. M. Kaminker, V. A. Nazarenko, L. F.
Saenko, L. M. Smorotritskii, G. I. Kharkevich, V. A. Knyaz’kov.
Ram 71 NP B 28,566: B. Ram, W. Williams.
Weinberg 58 PR 112,1375:S . Weinberg.
Wilkinson 71 PRL 26,1127: D. H. Wilkinson, D. E. Alburger.
Wolfenstein 64 PL 13,562: L. Wolfenstein.
Pilkuhn
Ref. p. 141 1.2 Particle properties : General formulas for decay and resonances 9
1.2 General formulas for decays and resonances
1.2.1 Decay angular distributions, phases, and penetration factors
For the two-body decay of a resonance d in its rest frame, Eq. (4) in 1.1.5 simplifies to
pdQ
dLips = -, mr,,=- 16,pG A; -.il dco~QI~~~,&m 9, d12 >
167c’sf
12 L 2
where p = 2s;$A*(sr 2, rnf , rnz) is the decay momentum, 1/slz is the invariant mass of particles 1 and 2 ‘),1 , and A2
are their helicities, M is the magnetic quantum number of d (averagingo ver M is unnecessary)9, is the angle between
the spin quantization axis of d and the momentum of particle 1, and cpi s a possible rotation angle around the spin
quantization axis (counted for example from the collision plane of the reaction ab --f cd). For identical particles in
the final state (x’+yy), the upper limit of integration in (1) is 0. The angular dependenceo f T is (S = spin of d):
TMIIA*(Q, rp) = (ghh 9, --VI T,Ul, 12)
2S+l 3
=- ei’p(M-“)d&,(9)T s(ll, a,),
4a
( 1
d&,,(9) = (exp( -iSJ~‘)),,. 1= ,I1- a2
Tab. 1. The functions d&,,(Q) for S = 3 ... 3. The sign convention of [Rose 57j is chosen, and dkMZ= d!,.-,
=(- l)M-M’dM,M
M’
\M +g
++ cos3+e -3+ coszL2 e sin’2 0 3* c0s +e sin’ &e -sin3 ‘2e
t-3 3fc0s2$8sin@ cos +e(i - 3 sin’+e) sin )e(i - 3 cos2:e) 3* c0sQ 2 sin221 8
-f 1 33c0s3esin238 1 sin )e(3 c0s2 +e - 1) I cos +e(l - 3 sin2)Q) I -3fc0s2@sin+tI I
-$ +sin3@ 33 cosle 2 sin’l2e 3* cos2~0 sinie ~0s~ ‘28
M M’ 2 1 0 -1 -2
2 c0s4”e 2 -2cos3162 sin9 2 6f cos2“2 0 sin”02 -2c0s’es2in 31e 2 sin4128
1 2 c0s31e 2 sin128 ~0~~fe(c0s~+e-3sin~~e) +sine~ose sin’te(3 cos2$e - iin +e)
0 1 6”.cos23esin2~81~ssinecose / 3(3cosz@-1) / +sinecose
M’
\M 2 23 T1 -4
I -5*c4s I 10+c3s2 - lOfC%3
I
; 5”c4s c3(c2, - 4s2) 2%c2(3s3- 2~‘) 2+s2c(3c2- 2s’) s3(s2- 4c2) 5+cs4
3 lOWs2 2fsc2(2cZ- 3s2) c(c” + 3s4- 6~‘s’) ~(6s’~’ - s4- 3~“) C~COS:B,S~S~&B
The necessaryd -functions are given in Tab. 1. 17’1in’ (1) is in fact independent of rp.T he decay angular distribution
for polarization density matrix 4 of d is (see[ Pilkuhn 671,r ef. 1.1.6):
WA cp)= c @LM,efM @,cp), Jdco&dqW=l, (3)
MM’
‘) 1 is definedi n Eq. (12)i n 1.1.1a, nd s in Eq. (5) in 1.1.5.
Pilkuhn
IO 1.2 Teilcheneigenschaften: Allgemeine Formeln fiir Zerfall und Resonanzen [Lit. S. 14
mF,, = p(32n2sf2)-’ c ITsO,,, A,)[*, since s d cos9(2S+ 1) lDl* = 2. (5)
1112
When particle d has spin $, one normally puts Q= f( 1 + P’+c ?)w, here P’ist he “polarization vector”. Ts is real except
in weak and electromagneticd ecays,w here it has the elastic scattering phase of the final state. For the decay into
two spinless particles. the simplest Lorcntx-invariant matrix elements are given in Tab. 2. For arbitrary spin S,
one has (s = s,*)
T,, (.%9 , (P)= (6)
Tab. 2. Matrix element T, width F and suppressionf actor N for the decay of a resonanceo f spin S into two spinless
particles. E,, is the tensor delined in Eq. (11) in 1.1.1.O nly G, is dimensionless
S Ts mF (Breit-Wigner) N
G
(47+ Go -p-f 1
z(332P 3 s -f 1 + R*p*
G4:X& ISP s s -f 1 + R2p2/3 + R4p4/9
i
It is noted that T gives rise to the “threshold behaviour” pzsc’ of F. This behaviour is reliable only for small p.
For large p (i.e. for p % p,,.,J, F certainly does not diverge like p *‘+’ . Several moditications of F can be used here.
A simple modification adopted from potential theory (R X range of potential) is to put
mF = ntF (Breit-Wigner)/N, N = R*p* Ih”‘(Rp)l* ,
with N given in the last column ofTab. 2. It must be noted that this may changet he coupling constant considerably.
For example.f or a p-wave resonanceo f given F(m), Gf/47r will be twice as large for R*p*(m) = 1 as for R*p*(m) = 0.
This ambiguity is very pronounced for the A resonance( see 1.4.7a).W hen the decay products carry spin, one uses
N(L). where L is orbital angular momentum. See also [von Hippel 721.
For S = 1, the angular distribution for decaysi nto two spinlessp articles is determined by 3 independent and real
parameterso f the density matrix e. For the spin quantization axis of d in the production plane of reaction ab+cd,
one finds
1-A
M’(9.p ) = 7, A=(p,,-e,,)(l -3cos29)+31/2Rep,,sin29+3e,~-,sin29cos2~. (7)
For S = 2. the corresponding distribution is [Dalitz 663
W(,9,c p)= -/& {3e,,(cos*9 - 4)” + e,, sin*29 + e2* sin49 - 2A, sin29 coscp
(8)
-4A, sin29cos2cp+ 4Ree,,-, sin39cos9cos3cp+ e2.-* sin49cos4cp},
A, = Reel, sin*9 +flRep,,(cos*S- f), A,=Q,.-~ cos*S-fiRep,,(cos*S-f). (9)
For pseudoscalara nd vector meson exchangea lone, e2* = A, = A, = e2,-* = 0.
Dcmrs in jlighr. The lifetime F,$’ is F-‘I&,/m due to the time dilatation. The phase spacee lement entering
Eq. (1) can be transformed to d E, ,,,h:
pdcos9 dE,,,,
dLips(s,,;P,,P,)= 8nP-t = -
Plnh = @;h - %2)+ .
‘12 8~~1oh’
For spinlesso r unpolarized particles d. ZITl* is independent of both 9 and q. In thesec ases,t he decay distribution
is also flat in Ellah.a s d Lips/d Ellah is independent of E, ,oha ccording to (10). Elloh is related to the emission angle
Qlnto, f particle 1 as follows:
(11)
Pilkuhn