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SLAGPm- 1~~48-
EtiCTROPRODUCTIONOF NUCLEONRESONANCES* J. D. Bjorken+ and J. D. Wale&a*
ABSTRACT
We compute the differential cross section for the process e+p+e+p (^) R
where pR^ is^ a^ nucleon^ resonance^ characterized^ by^ parity,^ spin^ J, a- and^ mass^ 53. The^ two^ inelastic^ form^ factors^ describing^ this^ cross section are expressed in terms of three smplitudes characterizing the (p,p,) electromagnetic vertex. The^ kinematic^ and^ analytic^ structure of these three amplitudes as a function of q2 are discussed. The case of the 33 resonance is discussed in some detail.
*Supported in part by the U. S.^ Atomic^ Energy^ Commission^ and^ in^ part by the Air Force Office of Scientific Research Grant m<6%)-1389. ',- _,' / t Stanford Linear Accelerator Center, Stanford University, Stanford, California.
- (^) A. P. Sloan Foundation Fellow, Department of Physics, Stanford Uni- versity, Stanford, California
-l-
I. INTRODUCTION
The study of inelastic electron-proton scattering is likely to in- crease in importance as a probe of the structure of the nucleon as higher energies and momentum transfers become available. Furthermore, from both the experimental and theoretical standpoint, excitation of resonant states (isobars) of the nucleon will be of particular interest. We here^ review and extend (1,2,2) the phenomenological description of such a process, giving the general vertex and differential cross section for e-+ p +e + p (^) R' where pR is^ a^ nucleon^ resonance^ characterized^ by^ parity^ nR,^ spin^ J,^ and mass M. We find^ the^ snalogue^ of^ the^ description^ of^ elastic^ e-p^ scattering by't?he Rosenbluth formula; our^ main^ result^ is^ the^ following^ expression for the cross section, where^ only^ the^ final^ electron^ is^ detected,^ and where the initial particles are unpolarized
da
dRI =4s2 sin4 g 1'26m sin2 '
1
92+Etap
lab (^) C P 2q*2^ m
Here E and 19 are incident electron energy and scattering angle, and M and m are isobar and nucleon masses, respectively. q2.= 4s~' sin2^ p'^ 1s^ the invariant four-momentum transfer, while q2 is^ the^ magnitude^ of^ the^ three- momentum transfer from the electron in the isobar rest frame:
9 *2^ =q2+-^1
4M"
I (1.2)
t:e know from the Bessel functions involved fcW (qR)L for qR < L where L = J + '2 depending on the parity and R is the radius of the target. On the other hand, kinematics [Eq. (1.2)] tells us
9” ’_^ g^ (M-m) Taking (M-m) N 2uL as is roughly the case experimentally (cl is the pion mass), we find
R5.$-^1 in order that threshold behavior still persist for spacelike q2. This is aa- rather small interaction radius. For the normal parity transitions, we find sn additional relation between fc and- f, (^) - valid near threshold, which is an additional test on &he spin-parity assignment and the assumption of dominance of the thres- hold behavior., This relation
Ifc I’^ z^ g+;^ q. lf+)2+ If- 1' (^) - q*
(1.6)
is well-known in nuclear physics. In^ particular^ it^ is^ the^ relation^ which allows one to get photon lifetimes for electric transitions from Coulomb excitation. The rest of this paper is devoted to the details of deriving the cross section and to discussion, as^ best^ as^ we^ can^ on^ general^ grounds^ alone,^ of
-Mwhat^ behavior^ might^ be^ expected^ of^ the^ form^ factors.
(1.8)
_-. -4-
In Section II, we write down the general electromagnetic vertex function between isobar and the nucleon. Two representations are used. The first is the Jacob-Wick (4) helicity representation, written down in the isobar rest frame. In the second we explicitly describe the isobar by a generalized spinor wave function (2) ly**c( 'r 3-k (^) (pi= 1,...,4; r = 1,...4) (1.10)
In each instance three form factors are involved.
.a- In^ Section^ III,^ we^ square^ the^ matrix^ element^ and^ sum over^ spins^ to obtain the cross section, using again both the helicity and spinor wave function methods. The^ connection^ between^ the^ two^ approaches^ is^ then^. ._ established. In Section IV, the threshold behavior of the form factors is derived. In Section V, we review the analytic properties of the form factors expected. from Feynman diagram considerations. Unlike elastic scattering, complex singularities appear and the form factors need not be real. Indeed in certain circumstances the phase of the form factor is determined by the Watson final-state theorem, and we discuss how this comes about. Finally, in Section VI, we summarize briefly implications of some models for the P3+ resonance,^ in^ particular^ that^ of^ Fubini,^ Nambu and^ Wataghin^ (6)^ and ofGourdin and Salin (I).
Helicity Analysis In the rest frame of the isobar the quantity we want to study is
< fiRJM Jp(0)I Is* X > where X is the helicity of the initial nucleon and q* = L = (^) VP is its momen- turn. We now want to use angular momentum conservation and the fact that
kJ(0)^ transforms^ as^ a^ vector^ under^ rotation^ while^ Jo(O)^ transforms^ as^ a scalar. Our final state is already an eigenstate of angular momentum. The problem is therefore to expand the initial nucleon state in eigenstates of angular momentum. From the work of Jacob and Wick, one knows immediately how to do this. The basic theorem is that
One gets the appropriate energy-angular momentum-helicity eigenstate by integrating the momentum eigenstate over solid angles and using as a weighting function the rotation matrices (^) BJ".ml ( We follow the angular momentum notation of Edmonds (g).) The final9 (^) q in the argument of the 8' (^) mxJ^ functions^ is^ merely^ a^ definition^ of^ the^ overall^ phase^ of^ the^ states (w eiW (^) q) as discussed in Jacob and Wick. The completeness'properties of the (^) Q' (^) mhJ allow us to invert this relation
(2.1)
(2.2)
and give us the needed result. That is
< .R’j+,)o,(~~> =
It is convenient at this point to introduce eigenstates of parity. From the work of Jacob and Wick the parity operator z acting on the nucleon state gives h fi I qjmh^ >^ =^ (-l)j-+(qjm^ -^ X >^. We can therefore introduce the parity eigenstates
Iqfijm^ >^ = 1J-F- Llqjmh^ >^ +^ (-l)j-+lqjm^ -X> 1 a-- The problem is therefore reduced to studying the matrix elements
where only the appropriate value of n can contribute. We can^ now use^ the Wigner-Eckart Theorem to extract the M dependence of the matrix elements.=
< XRJMzP(0)I Iqnjm >
Since in this case j = J,^ J+l^ there^ are^ three^ independent^ reduced^ matrix elements which are each functions of 1~1 = q*. We also have for the fourth component < xRJ ItJ,(O)ll (^) sd >
and only get a contribution here if J = j. There^ is,^ however,^ still^ one
(2.6 >
relation between these four reduced matrix elements which comes from the
- _-. -8 -
The continuity equation then simply eliminates f,
fo = + fc 9
and the transformation has the further property that
Spinor Wave Function Analysis
As an alternative to the helicity description given above, we may write the vertex function in terms of a general spinor wave function describing a particle of spin J. This wave function l l +J-+ (P,h) (r = I,..., 4; pa = 1, 2, 3, 4)
is-the generalization of the Dirac spinor for J = 4 (where there are no indices p> and the Rarita-Schwinger wave function for spin 3/2. It is
- Symmetric under any permutation of indices %.
- Zero upon contraction of any pair of indices pa.
- A solution of the Dirac equation
-w
(iyP + m)rs Yp .=,-+ -
"Orthogonal'^ to^ ycL^ and^ PI1;
Normalized ’ l l pJ-’ 2 (P,h) YPl l ,-$ (PJ) = 1
Our task is to reconstruct the electromagnetic vertex function in this language in analogy to the conventional (11)- relativistic treatment of, say, the proton vertex. From the helicity description we know that there will be three independent form factors; our^ choice^ of^ spinor^ covariants will be motivated (^) ---ex post facto. The choice we do make has the convenient property that cross terms between the different form factors vanish when the amplitude is squared and spins are summed in constructing the cross section. So without further ado, we write for the general form of the vertex function for normal parity transitions l/2+ -9 3/2-, 5/2+,...
9-1 (-P*q 1 -E-P s”> g,h2> x
- 2ie (^) p 1 cq3ypcxqpSy g2(q2) +M q Y'SY, g,(s') + g3(q (^1) (. where the pseudovector SCI is defined by SP = iepvha pqcv x u :
i UPA
For abnormal parity transitions the only change that need be made is the replacement
@,x) -+ Ys u(w)
(2.W
*_ -.-m
III. CROSSSECTIONS
The cross sections for the process illustrated in Fig. 1 can be written in standard fashion as (l.2) da =2# (^) --Wd3p' 1 IV 2E' q4 kLv^ (3.1)
where
NCLV= - 3 Tr y,(m$ - iyp) y,(m$ - (^) irp') (3.2) = (^2) IP,P; + P,PL - ~pv(P-p’+ m2) 1
p and p' are the initisl and final lepton four-momenta, m is the nu‘cleon mass and rnt is the lepton mass. (^) The covariant tensor WCLVis given by
WI.Lv = (2430 c c S(4)(p-p'-q) < PIJ,(o)(P'> < P'IJI-,(o)IP > (E) initial final (^) (3.3) where L? is the normalization volume, E is the energy of the target,
initialc^ indicates^ an^ average^ over^ nuclear^ orientations.^ From^ general considerations of covariance, parity conservation, and current conservation, 4J.wclV = Wllv%^ =^0 , the tensor Wyv is known to have the form (13)
WPV = w1(q2,q*p) Epv - I&! ( s2 J
The cross section in the laboratory frame is given by
- (^) I (^) -rVp (^) II p' (^) I case-2m. w (3.6)
- EE' + lpl[p'ldU COB 8 +mi W 3 If we neglect the lepton mass this formula simplifies d2a (^) = -402e2 - cos2e d.Q'dE' q4 m 2
- ~l(si?,q-P) tan se^5 1 (3.7)
a q2 =^ ~E'C^ sin^ 2ez
Knowledge of WPV also^ gives^ us^ the^ total^ photoabsorption^ cross^ section
- ._- iy = $ 1 VW ep Wclv eV = wWI(O, - k*P) PO
where the kinematics are illustrated in Fig. 2. We now proceed to calculate the covariant tensor WP-V and^ then the cross sections. We must evaluate d3P' WPV z&f- E' @(P-PI-q) 3 1 1 <^ fiRJMIJ,$O)(qA^ >^ <^ qhIJv(0)IflRJM^ > AM We see that we can take out d3P''r^ S4(P-P'-q)m2^ as^ a^ factor^ if the final state is an isobar and we .define d3P' Wpv =- El S4(P-PI-q) m2Tclv (3JO)
_- -^14 -
These equations give one the two necessary relations and we can solve to get
T =A^22 f ( 2 2 *4 c^2 Q I^ I^ +-$ (If+l’+^ If-13)
Inserting our expressions for W1 and W2 [c.f. Eqs. (3.5), (3.10) and (3.13)] in the formula for the cross section, Eq. (3.7), we find
X (^) [-$ /%I2+ ($ +5 tan2ii) (lf+12+ lf-121]
This formula is our main result.' We see^ that^ one^ can^ only^ measure^ the combination (^) (lf+12 + If-l') in^ experiments^ where^ only^ the^ final^ electron is detected. The^ Coulomb^ and^ transverse^ form^ factors^ can^ be^ separated experimentally by doing experiments at fixed q2 and varying 8 or by looking at 8 = 180' where^ only^ the^ transverse^ form^ factors^ contribute. Taking the limit m + -, M/m +l^ reduces^ the^ above^ formulae^ to^ that usually used in analyzing electron excitation of nuclei. Frcm Eq. (3.8) we have for the photoabsorption cross section integrated over the isobar
_- - 16 -
resonance
s uy(co)aW^ =^
4tr2c
over resonance M;! -^ m2^ "^ (^ lf+12^ +^ lf-l~2)q2;o^
This expression gives us the tra:nsverse form factors evaluated at one momentum transfer, namely q2 = 0. This last relation can be used to give an approximate formula for the inelastic electron scattering cross section at small q2, the Weizgcker-Williams approximation. Keeping terms of order q2 in the last bracket in Eq. (3.18) we find
a.- (^) do = dS1 (^) lab q2+ 0
Wave Function Description We may also compute W 1 and W2 in terms of the form factors g 1 , g 2 and g 3 and obtain a connection between them and the magnitudes of fc and f,. We return to Eqs. (3.9) and (3.10) for the expression for T (^) pV’:
vp T^ pv = s^ L^ <^ PX~Jv(0)(P'x'^ >^ <^ P'V~JV(0)~PX^ >^ (3.21)
We choose to evaluate (^) T (^) ClV in the isobar rest frame to facilitate doing the spin sums. We first construct explicitly the state x, (^). ..cxJ p for the case of the spin aligned along some axis 8. Were the z-&s
The integrations and traces are now straightforward; cross terms between different gi rancel out, and after some calculation we find
ET P w e'=v 2(^ E+m)(q^ “(2J4!! km2 2J-3 (2J)!! I^ I
gl
We insert the factor (^) Q2 to take into account abnormal parity transitions,
in which case the expressions Eqs. (3.24) and (3.25) contain an extra factor
due to the 7 5 :
& $ $, g,...
- To liberate Eq. (3.25) from the isobar rest frame, we compare it with
Eq. (3.13) and identify
(2J+l)!: g 2 2J!! (^) I I 1
-w
I
In terms of the f's, we thus find
[If+12+If-J2] =v. @gpI@+lg212+($
I
We again caution the reader not to read off the threshold (low q") behavior of the cross section from these formulae; detailed discussion of this point is in Section IV.
_- -^20 -
