I. THÉORIE / THEORYCURRENT ALGEBRA, PCAC, ITS ANOMALY, AND THE TWO-PHOTON PROCESS (SOFT-PION PRODUCTION BY TWO PHOTONS)

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I. THÉORIE / THEORYCURRENT ALGEBRA, PCAC, ITS ANOMALY, AND THE TWO-PHOTON

PROCESS (SOFT-PION PRODUCTION BY TWO PHOTONS)

H. Terazawa

To cite this version:

H. Terazawa. I. THÉORIE / THEORYCURRENT ALGEBRA, PCAC, ITS ANOMALY, AND THE TWO-PHOTON PROCESS (SOFT-PION PRODUCTION BY TWO PHOTONS). Journal de Physique Colloques, 1974, 35 (C2), pp.C2-61-C2-67. �10.1051/jphyscol:1974209�. �jpa-00215519�

CURRENT ALGEBRA, PCAC, ITS ANOMALY, AND THE TWO-PHOTON PROCESS

(SOFT-PION PRODUCTION BY TWO PHOTONS) (*)

H. TERAZAWA

Stanford Linear Accelerator Center

Stanford University, Stanford, California 94305, USA and

The Rockefeller University, New York, New York 10021, USA (**)

Résumé. — Diverses applications de l'algèbre des courants, de PCAC (hypothèse du courant axial partiellement conservé) et de son anomalie à la production de hadrons par deux photons (y + y -»• hadrons) sont passées en revue. Ces applications incluent les processus (1) y + y -s- n°,

r/, t,' (ou X°) et 6 ; (2) y + y -> 3 n», n+ + n~ + TT°, et (3) y + y -* /w+ + rm~ (n = 1, 2, 3, ...), ' où tous les pions produits sont de faible énergie. Le comportement asymptotique des processus (4) y* + y* -> n* + n~ et (5) y* + y* -> n° pour les photons hautement virtuels est également discuté.

Abstract. — Various applications of the current algebra, PCAC (partially conserved axial- vector current hypothesis) and its anomaly to the hadron production by two photons y + y -»• ha- drons are reviewed. They include the processes (1) y + y -> n°, ;/, >/' (or X°) and e (2) y + y -»• 3 n°, n+ + n~ + n°, and (3) y + y -> nn+ + nn~ (n = 1, 2, 3, ...), where all produced pions are soft.

Asymptotic behavior of the processes (4) y* + y* -> n+ + n~ and (5) y* + y* -> n° for highly virtual photons is also discussed.

In this talk I shall review recent work on the appli- cation of the current algebra, PCAC and its anomaly to the hadron production by two photons. I shall talk about the following subjects :

y + y -» 7t°, rj, q' (or X°) and s (1)

y + y -> 3 n°, n+ + n~ + n° (2)

y + y -> rm+ + nn~ (n = 1, 2, 3, ...) (3)

y* + y* -> n+ + n~ (4)

y* + 7* -> 7i° (5) where all the produced pions are soft (in other words,

their energies are small compared to, say, 1 GeV), and y and y* denote an almost real photon and a highly virtual photon, respectively.

I shall try to be consistent in such a way that all the predictions for these processes are based on the following three assumptions and on nothing else :

(a) SU(2) x SU(2) current algebra [1]

5(x0) K ( . Y ) , ! / > ) ] = ie.dbe Ve„(Q) ô(x), etc. ;

(*) Work performed under the auspices of the U. S. Alomic Energy Commission and Atomic Energy Commission Contract No. AT (11-1).

(**) Permanent Address.

(b) « strong » PCAC [2], and (c) the possible exis- tence of its anomaly [3]

3,XM =

= /„ m2n <pK(x) ( + ~~ e.,,, f ( x ) F»(x) for n°),

• 1 6 7T /

where fn (~ 93 MeV) is the pion decay constant and S is the anomalous constant.

By the « strong PCAC » I mean that the divergence of the axial-vector current is dominated by a pion pole and that the extrapolation of amplitudes needed from the off-shell point at pi = 0 to the on-shell point 'at pi = ml is smooth so that the physical ampli- tude can be approximated well by the soft-pion amplitude calculated at the unphysical point where Pi = 0 [4].

In order to save our time. I shall skip rigorous derivation of the results without any exception but, instead, give a rough idea about how to derive them.

Therefore, for those who are interested in the more details, I strongly recommend that they refer to the original papers or my review article [5] which will appear in the Reviews of Modern Physics soon.

Now let me start with the first subject.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974209

C2-62 H. TERAZAWA

1. y

+

tion of no by colliding beams, e'+e-+e'+e-+no, According to Bell-Jackiw and Adler [3], the anomaly was first proposed by F. E. Low [6] in 1960 as an exists and the low-energy theorem

experiment for measuring the life time of no. In the

equivalent-photon approximation the total cross sec- ² S

F(0, 0, m,) E F(0, 0, 0) = - ---

tion for this process is proportional to the decay 2 n 2 f n width Trio,,, as follows :

holds. The anomalous constant can be calculated bv 2 E the triangle diagram (see Fig. lu), with a fermion

c e e - t e e r r o ^---

111," 111, quark loop to be

-

The latest expel-imental value [7] for the width is Trio,,, = 7.8 0.9 eV. 111 order to obtain a better accuracy by two-photon experiments, I urge to observe scattered leptons within a small angle so that both theoretical and experimental uncertainties may be eliminated. It seems fairly easy to obtain better values for the decay width of -+ yy. q' + y y , and ^{E -+} y y in this way. Why is the measurement of these decay widths important ? I shall give one reason here. I expect the next speaker, Dr. Brodsky, will present us another reason.

The vertex function of no yy is defined by

4 for the Han-Nanibu model and the three-

\

S =

for the original Cell-Mann-Zweig triplet

( '

The present experimental data shows

which strongly favors the former two models. The anomaly also affects the low energy theorem on the q ^-+ y y and 11' -+ y y vertices. However, the low energy theorem is less practical in these cases than in the case of no -+ y y because the extrapolation needed is much more demanding.

The coupling constant of c(700) and two photons is theoretically as interesting as the no + y y decay constant. In order to measure this, it is desirable to do

= i

5

I

I

The decay width is given by e'

+

+

+

I e 2 F ( o , 0 , d ) 1 2 ³ in which both scattered lepton momenta are measured

r n ~ + y y ^{= HI,} .

64 n t o find a missing mass o r the one of the type By using the LSZ reduction formula and the modified

PCAC including the anomalous term, one can obtaifi the Ward-Takahashi identity at P2 = 0 :

The equal-time commutator terms vanish because the ,electromagnetic current commutes with the neutral current A;. This identity can be transformed into the relation

in which both pion momenta are measured t o find a broad ^E (or o) enhancement in the invariant mass distribution of the pion pair. It should be noticed here that the ~ y y coupling constant can be directly

/ I

where G's are the form factors of the AVV vertex.

Now one can see that (b)

measured in the first type of experiments while what can be measured in the second type is the product of the ~ y y and E X + n - coupling constants but not the

~ y y coupling constant alone. Theoretical predictions for the product of these coupling constants and their consequences will be discussed by the next speaker.

For the ~ y y coupling constant we can play the same game as we just did for the no y y decay constant.

What we need is to replace PCAC by PCDC (par- tially conserved dilation current) [9]. Kleinert, Staun- ton and Weisz [lo] showed that if the ~(700) meson dominates the trace of energy momentum tensor

02,

then the ~ y y coupling constant vanishes in the soft- meson limit. However, Crewther [I I] and, indepen- dently, Chanowitz and Ellis [I21 have recently pointed out that the PCDC anomaly [13] affects the low energy theorem and that the ~ y y coupling constant does not vanish (see Fig. lb). Furthermore they have predicted the coupling constant defined by

where R is the asymptotic ratio of a(e+ e- ^-+ hadrons) to a(e+ e- + y + k i p ) a n d f , is defined by

From this result Chanowitz and Ellis [12] have estimated the ^{E -+} y y decay width to be

re,;, ^{z 0.2} R2 keV for 171, E 700 MeV and

.f, E 150 MeV .

This can be checked by two-photon experiments.

Next, I shall discuss the production of an odd number of soft pions by two real photons.

2. y + ^{y -+} 3 no, ^.n+

+

+

+

viously are not gauge-invariant (see Fig. 2). The conclusion of these various authors is the following : (1) Both of the amplitudes for y

+

+

+

+

+

lr,+ 70 7- ^lr+ _, ^{lrO lr- lr+ 71;O} y-

I

\ I \

lr+ lrO lr- '\,,; ; , : I : ; :

\ I / \ I f

\ 1 \ I /

\ I /

I' \ I

R ^-

Y Y

Y Y Y Y Y Y

and y -+ n+

+

+

due to the gauge invariance of the whole amplitude, where FRO,,, = F(0, 0, 0) and F,,,+,-,o is defined by the soft-pion production amplitude

The proof of (3) in the presence of the PCAC ano- maly has recently been given by Terent'ev [17] and Adler, Lee, Treiman and Zee [19] although the relation between no ⁺ y y and y ^-+ n f

+

+

With the corrected version of the amplitudes for y + ^{y +} 3 no and n+ + ^{7 1} + no, actual calculations of the cross sections fore + e ⁺ e

+

+

+

+

+

-

in the soft pion region where these soft-pion results should be tested. The result of Koberle 1241 shows, however, that the hard-pion cross section for these processes calculated from vector meson domi- nance may be large enough (-- cm2 for e + e + e + e + n f + n - + n O a t E = 1 GeVand

-

+

I shall now move to the production of even number of charged soft pions by two photons.

3. y + l l + n n t

+

C2-64 H. TERAZAWA

pion limit and in extrapolating the off-shell ampli- tude to the physical one. However, I will take gauge invariance and the Thomson limit for forward Comp- ton scattering as guiding principles to obtain a consis- tent result.

Let us start with the definition of the amplitude for y

+

+

The successive application of the PCAC hypothesis, the soft-pion technique, and the algebra of currents makes it possible to reduce a pion pair in the limit p, q ^-P 0. Repeating the same procedure ⁱ⁷ times, I

shall end up with

where V ; and A: are the strangeness conserving vector and axial-vector currents (their third compo- nents in the isospin space). Using the spectral repre- sentations of the propagators, one can obtain the following expression for the matrix element

We can clearly see that not only Lorentz covariance but also gauge invariance is maintained if and only if Weinberg's first sum rule [27]

is valid. Therefore I shall assume the validity of the sum rule hereafter. Thus I have. obtained the soft- pion theorem [28]

and

with F(0) = I , which gives a correct Thomson limit for n = 1 when k , and k2 vanish. This relation has also been obtained by Terent'ev [29] for n = 1 and

has been confirmed by Goble and Rosner [30] for n = 2 and in the limit of k, and k2 -+ 0.

Now we can apply this soft-pion theorem to various physical processes. For example,

and

We should notice that the cross section for two-pion- pair production is two orders of magnitude larger than that for n f n- no production -- low3' cm2 at s = (4 n ~ , ) ~ . More detailed and precise numerical results for n = 2 can be found in the paper by Goble and Rosner [30].

Within the equivalent-photon approximation we can estimate the cross section for e

+

+

+

+

sort pion = 3.5 x cm2 at E = 1 GeV .

The next subject is how to extend the soft-pion results obtained for real photons to those for virtual photons.

4. y*

+

+

+

As we have just seen, the PCAC hypothesis and the algebra of currents determine the amplitude for y *(kl)

+

+

k, + k2 = O and q, = q, = 0 . From this point we must extrapolate to the nearest physical point, namely, the production threshold q1 = q2 = (mn,O,O,O) and k,

+

In addition to k: and k;, there are two invariant variables available, namely, s = (k,

+

+

(see Fig. 3a)

as strictly as possible, we can measure the theoretically interesting quantity

with

~2 = - k2 - - k2

1 - 2

for arbitrary values of Q2. The differential cross section is proportional to [F(Q2)I2.

On the othe- hand, several years ago Das, Guralnik, Mathur, Low and Young [33] derived an expression for the electromagnetic mass difference of the pions based on the same assumptions as we have just made.

It is given by

m

in,. 2 - m:. ⁼

J

Combining these two things together, Yan [31] has arrived at a sum rule which looks as follows (see Fig. 36) :

3)

2 do(ee + eeni n-)] ^{l i 2}

m.+ - iw:, =

/

s+n,,,: [(known function)

Therefore, the electromagnetic mass difference can be determined by the process

This experiment is extremely interesting for the follow- ing reasons : (1) a measurement of the function F(Q2) is of great theoretical interest because it may answer such questions as the convergence of Wein- berg's second sum rule, the behavior of the spectral function of the axial-vector current, and so on.

(2) The origin of the pion mass difference is not known. Most people may believe that it is electro- magnetic. 1 think, however, that it is still an open and interesting question whether the pion mass difference is really and entirely electromagnetic or not.

Unfortunately. the effective cross section for large Q2 ( > 0.5 GeV2) is too small ^{( c -} 10-" cm2) to be measured i n the near future. Detailed calculations

of the cross section for this experiment can be found in the paper by Isaev and Khleskov [34].

Next, I shall talk about the asymptotic behavior of the no yy vertex function for large mass of virtual photons. It can be studied by the two-photon process

Let us go back to the definition of the vertex function

= i

1

/

f))

and

Several years ago, Cornwall [35] showed that the no y y vertex function approaches the limit

2 f, q2

F(q2, k2, m:) ^{+ -}

,

3 q k2

if there is no q-number Schwinger term, if the BJL theorem

P,Q fixed

is valid, and if the quark model for the space-space component of the equal-time current commutators

holds. Furthermore, Gross and Trieman [36] have predicted the scaling of F(q2, k2, m:)

q2 k2, m:) -+f(a)

in their scaling limit, q2 + GO with w = k2/q2 fixed, by assuming the gluon quark model for the light- cone current commutator [37]

and

Since these results are not only model-dependent but limited to the special kinematical region where

C2-66 H. TERAZAWA both q2 and k 2 are large with the ratio k2/q2 fixed,

it is desirable to find a less model-dependent and more widely applicable prediction for the asymptotic behavior of the vertex function. I have found that the vertex function, if it decreases at all, should decrease not slower than 114- q2 for any fixed values of k2 [38]. I have shown this by using the Schwartz inequality and unitarity only (see Fig. 4).

More detailed discussion on the consequences of this inequality can also be found in the recent paper by West [39].

2

,<C, *

I m " ? I Irn X I m

Y * Y * Y * Y* Y *

How does the PCAC anomaly affect the asymptotic behavior of the no y y vertex function ? To see this, let me remind you of the following expression [40] :

We can clearly see that, if the no y y vertex function decreases at all, a combination of the form factors of the AVV vertex should decrease as fast as (S/2n2) (q2

+

which is a very strong constraint on the form factors.

Although the effective cross section is small for large - q2 and - k2 [- cm2 for E = 2 GeV and - q2, - k2 > 1 GeV2], it seems easy to find whether the vertex function decreases or whether it scales.

In conclusion, I think that the asymptotic behavior of the inclusive process

y

+

in which both photons are highly virtual is theore-

+ 2 [- ⁺

+

fn 2 and completely determined by the algebra of bilocal

currents instead of the algebia of currents. I expect that Dr. Walsh will talk about this in great detail.

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and YOUNG, J. E.. P ~ J J S . Rev. Lett. 18 (1967) 759.

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DISCUSSION D. H. LYTH (Lancaster). - I want to ask a question

about the current algebra predictions for y y -+ n+ n-.

Are these predictions compatible with the Thomson limit which should hold for zero mass pions at thresh- old ? Your function F(q2) giving the dependence on the photon mass does not seem to be equal to the pion form factor, as one would expect if the Born term (which generates the Thomson limit) were dominant.

H. TERAZAWA. ^- Yes, they are compatible with the Thomson limit. However, as I mentioned, the Thomson limit, gauge invariance, and Lorentz invariance are assumptions in my calculation rather than consequences. In order to obtain the results compatible with those, I have assumed the validity of the Weinberg's first sum rule. The function F(qZ) is different from the ordinary pion form factor. This means that, for large q2, the Born terms are not domi- nant. The function is supposed to account for all the complexity of strong interactions.