4 January 1996 

Physics Letters B 365 (1996) 264-274 

PHYSICS LETTERS B 

A note on the yy + 7~~7~’ reaction in the 1 /N expansion of 

/ITT 
Antonio Dobado a*‘, John Morales b,2 

a Departamentv de Ft’sica Tedrica, Universidad Complutense de Madrid, 28040 Madrid, Spain 
b Departamento de Fisica Tebrica, Universidad Authoma de Madrid, 28049 Madrid, Spain 

Received 14 September I995 

Editor: R. Gatto 

Abstract 

In this work we present the results of a complete calculation of the yy --) ?r”~‘:’ amplitude to leading order in the large 
N approximation (N being the number of Goldstone bosons) up to order r&/F’. The amplitude turns to be proportional 

to that of rr’rr- -+ -rrarr”. In spite of the fact that this factorization property cannot hold in general (as it was recently 
pointed out by Morgan and Penington), it appears here since in the large N limit only the I = J = 0 channel contributes to 
the yy --f Z-~TT” reaction. Moreover it seems to be a reasonable approximation in this case since it is possible to reproduce, 

as a prediction, the experimental data starting from a one-parameter fit of the Z-T scattering data. 

1. Introduction 

In the last years a great deal of work has been devoted to the yy + n-‘7r” reaction. The reasons for that 

are the experimental data available from the the Crystal Ball Collaboration [ I] and that this process, being a 
pure finite one-loop effect, is a very good test for the modern Chiral Perturbation Theory ( xPT) [ 21 which 

successfully describes many low-energy hadronic reactions. However, for yy -+ r’n-‘, the one-loop prediction 
[ 31 disagrees with the experimental data even near threshold. This is due to the pion rescattering which is very 

strong in the I = J = 0 channel. Nevertheless it has been shown that the use of dispersion relation techniques 
[ 41 can provide a good description of this reaction and reconcile data with xPT [5 J. More recently it has been 
shown that a two loop-computation produces a cross-section that agrees rather well with the available data and 
with dispersion theoretical calculations even substantially above threshold [ 61. 

Another completely different approach that has been proposed for this reaction by Im [7] is the so-called 
large N expansion (N being the number of Nambu-Goldstone bosons) . This method was first used for elastic 

pion scattering in Ref. [ 81 in the context of the Linear Sigma Model (LSM) and in Ref. [ 91 in the Non Linear 

’ E-mail: dobado@cemvm.cem.ch. 

2 On leave of absence from Centro Intemacional de Fisica, Colombia. 

0370-2693/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved 

SSDI 0370-2693(95)01240-O 



A. Dobado, J. Morales/Physics Letters B 365 (19%) 264-274 265 

Sigma Model (NLSM) . Pion mass effects have also been included in Ref. [ lo] for the LSM and in Ref. [ 111 
for the NLSM. In this last work it was shown how it is possible to fit the J = 0 pion scattering phase-shifts up 
to very high energies. 

Nevertheless Im’s work has been strongly criticised by Pennington and Morgan in Ref. [ 121 because some 
of his assumptions are only true for point interactions which is not the case of pions. Since Im’s computation 
is not in fact a complete large N computation, these assumptions cannot be supported and thus we consider 
Pennington and Morgan criticisms appropriate. In this note we clarify this issue by making a complete large 
N computation of the yy + ?r’z-’ reaction. Our main conclusion is that at the end the original Im’s result is 
correct at leading order of the large N expansion and to first order in mi/F2 so both Pennington and Morgan on 
one side as Im on the other, are simultaneously right. In addition we will show how the large N approximation 
provides a very good description of the Crystal Ball experimental data. 

2. The chiral lagrangian in standard S3 coordinates 

As it is well known, the low-energy interactions of pions and photons can be described by a I/( 1 )em gauged 
NLSM in the context of modem xPT. This NLSM is based in the coset space S(l(2)~ x SI!/(~)R/SU( 2)L+R = 
0( 4)/0( 3) = S’ = SU(2). In order to define the large N limit we will extend the coset space to 0( N + 
1 )/O(N) = SN and we will define the corresponding lagrangian as follows. First we start from the LSM 

L: = iafi4Ta’L4 + dfFm2a (1) 

where 4T = (7rI,d,,9r2,7j2 ,.., rk, ?rL, TN, a) which is 0( N + 1) invariant if it was not for the last piece that 
explicitly breaks this symmetry to O(N) (with N = 2k + 1) . This contribution will provide the mass term for 
the pions and other interactions forced by the chiral symmetry. With 7~ = ?rN the above lagrangian can be 
written as 

C = ;a,ruabfJ + ~ap7T;abP + $a,r”a%o + ;afigapg + fiFm2a 

with a = 1,2,.., k, or, in terms of the complex charged fields q+ = (+’ + irr’O) /a, as 

C = a+pp:afi# + ~a,7T0afi~0 + ia,aafiu + fiFrn2r 

The electromagnetic interactions are included by introducing the corresponding covariant derivative 

apL’p* --+ Dp,(pu =: a& -I- ieA,c,o’ 

and adding the photon kinetic term 

(2) 

(3) 

(4) 

where Fpy = a,A, - apA,. Now we obtain the NLSM describing 
4T+ = NF* = fi to force the system to live in the SN sphere i.e. 

the low-energy pion dynamics by setting 

r2 $ a2 = NF2 (6) 

where d = Ci=, (ti* + 7~“‘~). Thus we have (T = (NF* - T*)‘/* 

NLSM becomes 
and the lagrangian of the U( 1 )em gauged 

, 
C, = a 

1 
cp+afip + La 

2 fi 
TOapTO + i (aP(D+q + (“+acLq + roaPno)L 

2 NF2 - 2cp+(p 
+ 

+e2 A*cp+cp + ieA,dkqip + h.c. - t Fpy Fpv (7) 



266 A. Dobado, J. Morales/ Physics L&tiers B 365 (1996) 264-274 

with (pT = ($0, (Pk) ,.., . 
This way to write the lagrangian describing the low-energy pion dynamics corresponds to a particular 

coordinate choice on the coset space SN but in principle any other parametrization connected with this one 
through an analytical change of coordinates on the coset manifold will yield the same S matrix elements (but 
not necessarily the same Green functions) [ 131. In particular, for the N = 3 case a very popular coordinate 
choice consists in defining the following SU( 2) = S’ matrix: 

,i“W“( I, 

x=eT (8) 

The change of coordinates cp’ = cp’( w), with i = 1,2,3 relating both parametrizations of the coset space is given 

by f,J = q54 + $7; with q54 = dm where +2 = Cl, @’ with4r= [$'lt~2,&>~41 = b%~~,~O,~l. 

The above lagrangian written in terms of these new coordinates reads 

DJ = d,I: + ie [Q, C] A,, Q = diag (2/3, - l/3) (10) 

As it was commented above, both lagrangians yield the same S matrix elements and the same observables like 
phase-shifts, cross sections, etc. However, the Green functions could be different. Indeed the computations can 
be much easier using one set of coordinates than another since the Feynman rules and the diagrams contributing 
to some given process will be different in general. In particular, the computation of the one-loop yy + ?r”no 
amplitude is simpler using the #J standard S’ coordinates than the 2 (chiral) ones, which were used in Ref. 

[31. 

3. The one-loop scattering amplitude 

The yy + TZ- transition amplitude is defined as 

A (YY + ?T”.?ro, rr+71--> = EIP (kl> E2v (k2) Mg+,+,- (11) 

where eIP, ~2~ and ki, k2 are the polarization vectors and the momenta of the two photons. 
The two one-loop diagrams for the process yy -+ TOT’ using standard S3 coordinates are shown in Fig. la. 

Note that when this computation is done with chiral coordinates there are two more Feynman diagrams [ 31. In 
our case, using dimensional regularization, the diagrams are given by 

[(! + k, j2 - m;] [(t - kA2 - ~$1 

Niy = =& (s - m;) J’ dl (2!+kd,W-Wv 

(F -,,I;) [(P_+k,)= -m;] [U- k2)= -.,$I 
+ [(k,,,d - Uw)] 

(12) 

with 



A. Dobado, J. Morales/ Physics Letters B 36s (19%) 264-274 267 

a) x ID( + crossing 

b’ x IllI + crossing 

‘)x y x + crossing 

d) 

Fig. I. Diagrams contributing to (a) the yy - ## reaction at one-loop level, (b) the tree level yy + x+r-, (c) the one-loop 

yy + r+r- and (d) the one-loop pion wave-function renormalization. 

It is not difficult to see that the above Nzy and Nj,, correspond respectively to the terms MLy + Mzy and 

ML,, + M$, of the second reference in Ref. [ 31. The complete matrix element is given by 

M,, = 
g,, (I’ - mi) - (2!+ k~ lp (21- k21Y 

(e2-m$) [(!+k~)~-m$] [(E-k212-mS] 
(14) 

which is proportional to the xPT lowest order ~T+C + 71.‘~’ amplitude given by (s - m$)/f$ 

As it is well known, the integral above is finite and therefore no renormalization is needed. This is because 

there is no four derivative term in the gauged NLSM contributing to yy + n-‘orr’. Therefore there is no any 
possible counterterm to absorb the divergences appearing in the one-loop computation and thus the consistency 
of ,yPT renders this one-loop amplitude completely finite. The result is given by 

M,, = & (s-m:) (&) (gpu - 2) [I + !${lnQ.-i~}~] 

From this amplitude one can calculate the total cross-section which is found to be 

m2 
(16) 

where 



268 A. Dobado, J. Morales/ Physics Letters B 365 (1996) 264-274 

(To (n-+6 + 77%0) = 32~ iNF2)’ cs -pm)2 
which is the lowest order T+T- + TOT’ scattering cross-section and 

(17) 

Thus, from the above equations one can realize that, as it was remarked in the second reference in Ref. [ 31, 

the lowest order GT+T- + TOT’ amplitude and cross-section factorize the one-loop yy -+ ?roro ones. 
From the diagrams in Fig. 1 it is also possible to compute the reaction yy + T+T-. However, in this case 

we have a tree level contribution (Fig. 1 b) and, in addition, we have to consider at the one-loop level (Fig. lc) 

the contribution coming from the pion wave-function renormalization (Fig. Id). In the reference frame where 

the photon polarization vectors EI , ~2 satisfy 

k, . cl = k, . l 2 = k2. el = k2. c2 = 0 

the amplitude at tree level is given by 

(19) 

pf . E]p- . E2 p- . E,p+ 62 

kl .P+ k2 .p+ 

where kl , k2 are the photon momenta and p+, p- are the VT+, w outgoing momenta. Now including the 
one-loop corrections. 

one-loop 
A( YY+“+“-) = 1&2NF2 

I ~2 (s + rni ln2 QT) 

l , . ,c2 _ p+ i”;;’ E2 _ p- ke;Pd_’ E2 
I . I 11 

with 

A0 (mi) = rni (A - lnm:) , A=~+log4?r-y+1+ln~2 

so that the full amplitude is 

A (YY + T+T-) = [d~;y_T+T_j +dy;;z'+_m,l 2, 

where the wave function renormalization constant is given by 

27r = 1 + ,(jTiNF2 A0 (47) 
so that we can write 

d (YY + lT+7i- I= 21e2 { ae1 62 - 
p+ . Elp- ‘E2 p- .e,p+ .E2 

k, .p+ - k2 .p+ 

where a is given by 

a(s) = I - 

(21) 

(22) 

(23) 

(24) 

(25) 

(26) 



A. Dobado, J. Morales/ Physics Letters B 365 (I 9%) 264-274 

a) 

269 

b, x xD( + crossing 

Fig. 2. (a) Diagrams contributing to the ?r - n- elastic scattering at leading order in the I/N expansion up to order mi/F’. The large 

black dot represents the pion coupling proportional to n$. (b) Diagrams contributing to the my -+ T”# reaction to the same level of 

approximation. The black square represents the addition of all the diagrams appearing in (a). 

4. The large N limit approximation 

In order to compute the yy + TOTO cross-section at leading order in the l/N expansion we go back to 
the lagrangian in Eq. (7). The pion scattering amplitudes to order l/N and the lowest order in ~$/fi were 
computed in Ref. [ 111. The contributing diagrams in this approximation are shown in Fig. 2a where the pion 
vertex also includes the infinite set of counter terms needed to renormalize the bare result. The renormalized 
amplitude can be written as 

Tahcd = &t&A (s) + &&dA (t) 

where 

1 
A (s) = - 

GR(% P) 
NF= 1 _ ?%&&T(s; p) 

with 

(27) 

s 

(47r)= F= 
lognz” 

p2 
(28) 

T(s;p) zz (29) 

The GR( s; p) and JR( S; ,s) are the generating functions of the counterterm couplings and they have a well 
defined dependence on the iu scale, in order to ensure the P independence of the whole amplitude. The isospin 
channels for the model here considered are defined as [ 141 

To = NA (s) + A (t) + A (u) 



210 

fi = A (t) - A (u) 

T2 = A (t) + A (u) 

A. Dobado, J. Morales/Physics Letfers B 365 (X9961264-274 

(30) 

and the partial waves are given by 

I 

J d (cost59 T, (s,cosO) PJ (cos6) (31) 
-I 

As it is well known, the requirement of unitarity strongly constraints the possible behaviour of these partial 
waves. In particular, they should have a cut along the positive real axis from the threshold to infinity. The 
physical amplitudes are the values just on the cut, and, in addition, the condition of elastic unitarity: 

Im a/J = 0JaiJ)’ (32) 

(where (T = VW) has to be satisfied in the physical region s = E2 + ie and E* > 4mi, where E is 
the total center of mass energy. This equation is exact for energies below the next four pion threshold but even 
beyond that point it is approximately valid. 

In the level of approximation considered in this work (large N limit and first order in rni/ F2), the dominant 
channel is the 1 = J = 0. As it can be expected the elastic unitarity condition is not exact but it can be shown 
that one has 

Ima00=~Ja0012+0 i 0 + O(&) (33) 

From the partial waves it is possible to define in the standard way the phase shifts S as 

t,J(S) = @%J(,y sin&(s)/c(s) (34) 

Now one can try to compare the results of our approximation with the experiment. Then one finds that the 
I = J = 0 pion scattering phase shifts can be fitted with the simple choice GR( s; p) = 1 and JR( S; p) = 1 for 
p = 775 MeV, as it is shown in Fig. 3. Concerning the I = 0, J = 2 channel (which is the other relevant for 
the yy + TOT’ reaction), the prediction of the large N approximation is that it is suppressed at leading order 
in the 1 /N expansion. In any case, one can use the above equations with the p value fitted for the ~00 channel. 
The result, which is in fact a prediction, is also shown in Fig. 3 and, as it can be seen, the agreement with the 
experimental data is pretty good. 

At the same level of approximation, the diagrams contributing to the yy + nor0 reaction can be found in 
Fig. 2b since it is not difficult to see that any other diagram is suppressed by extra l/N or m$/F2 powers. 
Thus the amplitude corresponding to this reaction can be written as 

M,, = e2 (2g,J(s) - I(k1k2)~~ - l’Uqk2),,) NA(s) +0(1/N) +OU$‘) (35) 

where 

I(s)= dr .I 1 

(l-m$)[(l-p~ -p2)2-m$l 

l(k,k2jp,, = 
s 

di 
(21+ kl ),(Z - k2)v 

(~-m~)[(I+k~)‘-m~lI(I-~2)2-m~l 

I’( 1?~k2)~cr, = 
s 

da 
(21- kr ),(21+ k2)u 

(1-m~)I(1+k2)2-m~l[(1-kkr)2-m~l 
(36) 



A. Dobado. J. Morales/ Physics Letters B 365 (1996) 264-274 271 

Fig. 3. Phase shifts for J = 0 elastic pion scattering. The dashed line represents the fit done with the leading order of the I /N expansion 

up to the order m%/F’ for the I = 0 channel and the prediction for the I = 2 channel. The continuous lines are the phase shifts as 

obtained from the standard one-loop ,yPT (see [ I I ] for the details). The experimental data corresponds to: A Ref. [ 151, 0 Ref. [ 161, 

0 Ref. [17], 0 Ref. [18], v Ref. [19],* Ref. [20], x Ref. [21], l Ref. [22]. 

The last two integrals have the same value, so that we can write 

M,, = 2e2NA(s) 
s 

df 
g,, (j2 - m;) - (21+ k~), (21- k), 

(e2 - m$> [( l+kl)‘-rni ] [V - kd2 - 4-j 
+O(1/N)+0(C$~2) (37) 

Note that this is the same integral that appears in the one-loop computation in Eq. ( 14). Therefore we also find, 
in the approximation here considered, the factorization property that we found there. In particular, Eq. ( 16) 
can be applied again although now the lowest order T+z-- 4 TO%-’ cross-section has to be replaced by the 
cross-section obtained from Eq. (28), which is given by 

r7 (7r+n- ---f 7i-0770) IA( 
= 32~ (NF2)2 s 

(38) 

using this formula, with the previously fitted value for ,U in the I = J = 0 elastic 
are given in Fig. 4. For completeness we also show the results obtained when the 

The results obtained by 
pion scattering channel, 
J = 0, I = 2 channel contribution is included by using the exact formula 

u (7f-t7i- + DOT’“> = 87T 
9(S - 4m2,) 

sin2( So0 - 620) 

together with Eq. ( 16) and the phase shifts computed in the large N approximation described above which 
are shown in Fig. 3. As it can be seen, the effect of the I = 2 channel is not very large, since it is not so 
strongly interacting as the I = 0. In any case our computations are compatible with the experimental data in 
both cases (with and without the introduction of the I = 2 channel) if one takes into account the large error 
bars. Thus, the large N approximation makes it possible to fit all the data shown in Fig. 3 and Fig. 4 with just 
one parameter, namely the scale ,u = 775 MeV. 



272 A. Dobado, J. Morales/Physics Letters B 365 (1996) 264-274 

i 

Fig. 4. Cross section for the process yy - T~‘T” with 1 cosHj 5 0.8. The continuous line represents the prediction of leading order of 
the 1 /N expansion up to the order m$/F’. The dashed line represents the same but including the contribution of the I = 2 channel. The 

experimental data comes from the Crystal Ball Collaboration [ I 1. 

5. Discussion 

In some sense the aim of this work is to clarify to what extent the above mentioned factorization relation 
between the yy + r”rro and the n-+~ + z-‘T’ amplitudes occurs beyond the one-loop approximation. Note 
that, as it was stressed by Pennington and Morgan in Ref. [ 121, the computation of the exact amplitude will 
require the knowledge of the n-+7~- + n-On-’ amplitude not only on-shell but also off-shell in order to make 
loops. In addition, the partial waves of the first reaction behave at threshold as sJ12( s - 4~2;) J/2 whereas the 

second goes as (S - 4mi) J and therefore the factorization formula cannot be correct. Moreover, from elastic 

unitarity (at the lowest order in the electromagnetic interactions), we have the relation 

Im3* = cr3**t IJ IJ IJ (40) 

involving the yy + n-On-’ and the VTST partial wave (we have used the standard definitions which can be found 
for instance in Ref. [ 51). In this reference it was shown how it is possible to use dispersion relations to 

improve the unitary behaviour of one-loop xPT, both for elastic pion scattering and yy ---f TOGTO. As a result, 

an excellent agreement with the data can be found in both cases. These dispersion relations have to be applied 
to each channel independently. Therefore, as the total amplitude is a linear combination, with well defined 
coefficients, of the different channel partial waves, we arrive again to the conclusion that the factorization 

relation cannot be hold in general. 
Therefore, for many different reasons, we known that the factorization formula which appears as a result of 

the large N approximation (as well as the original one-loop) cannot be true for the exact amplitudes. Then the 
natural question arises: Is there anything wrong with the large N approximation? and in that case: Why does it 

reproduce so well the experimental data? 
Of course the answer to the first question is no. The large N approximation is a well defined expansion in 

the l/N parameter and it is perfectly right. In particular the computations done by using this approximation 
satisfy the current constrains of any Quantum Field Theory like unitary, analyticity, crossing and so on, modulo 
higher order corrections in the expansion parameter, i.e. 1 /N (see Eq. (33) ), exactly as it happens in any other 
perturbative expansion. The reason why it gives rise to the factorization formula is the following: according to 



A. Dobado, J. Morales/ Physics Letters B 365 (1996) 264-274 273 

Eq. (27) the A function, which in principle depends on s, t and u, is only a function of s at leading order in 
the l/N expansion. In addition, from Eq. (30) we see that, strictly speaking, only the I = 0 channel contributes 
in this approximation. In other words we can say that to leading order in the large N expansion the only 
non-zero channel is I = J = 0. Then the n-+7rTT- -+ TO&’ amplitude only depends on s and only has an isoscalar 
component. This explains why all the previous mentioned objections to the factorization formula do not apply 
to this case as it happens in the general case where more than one channel contributes to the reaction. Therefore 
there is no inconsistency in the factorization formula obtained in the large N limit. However, this formula 
should be corrected by higher order computations (both in l/N and in mi/F') which will also include other 
channels different from the I = J = 0. In that case the factorization formula is not justified any more. 

Of course one can ask how good the large N approximation to leading order in 1 /N and in rni/ F* can be, 
since it only includes the I = J = 0 channel contribution. The answer can be found in Fig. 4. There we see that 
the agreement with the experimental data is good, and in fact it considerably improves the one-loop computation 
(the two-loop computation in Ref. [ 51 also reproduces well the data but depends on more parameters) 

6. Conclusions 

The main results of our work are the following: 
In order to properly define the large N expansion for two flavor xPT we have parametrized the coset space 

S3 -+ SN by using standard coordinates instead of exponential (chiral) coordinates. We have reobtained the 
well known one-loop results for the yy --f ~TIT reactions with this parametrization in a much simpler way, thus 
illustrating, with a very non-trivial example, the S matrix invariance under reparametrizations of the NLSM. 

We have computed the yy ---f r”7ro scattering amplitude to leading order in the large N approximation up to 
order mt/F2. In this approximation this amplitude turns out to be proportional to that of V+V- 4 r”rro (the 
factorization formula) according to Im’s conjecture (but in principle only for small rni/F') . 

Using a one-parameter fit of the elastic pion scattering in the I = J = 0 channel obtained in the same 
approximation we reproduce, as a prediction, both the I = 2, J = 0 pion phase shift and the yy -+ 7r”ro 
cross-section experimental data (see Figs. 1 and 2). 

From very general considerations it can be shown that the factorization formula cannot be exact and this 
was the basis for the Morgan and Pennington criticism to Im’s results. However Im’s result is correct in the 
framework of the large N approximation (for small mi/F2) since only the I = J = 0 channel contributes to 
the leading order in the l/N expansion. As a matter of fact, as the yy + 7r”n-’ is dominated by this channel, 
the factorization formula yields a good approximation as it was suggested by Donoghue et al. in Ref. [ 31. This 
would not be the case if, for example, the p resonance played an important role in the reaction, since it cannot 
be reproduced by the leading order large N approximation [ 111. 

In summary, we consider that the large N approach to xPT is a very interesting complementary alternative 
to the standard one-loop computations which can be useful in many cases in order to describe the relevant 
physics of the considered phenomena. This is particularly true in those cases where the I = J = 0 is dominant 
as it happens in the Higgs physics. 

Acknowledgments 

This work has been supported in part by the Ministerio de Education y Ciencia (Spain) (CICYT AEN93- 
0776) and COLCIENCIAS (Colombia). The authors thank F. Cornet, J.F. Donoghue, J. Gasser and J.R. Pelaez 
for useful information and discussions. 



274 A. Dobado, J. Morales/Physics Letters B 365 (1996) 264-274 

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