Pion electromagnetic polarizabilities and quarks

Volume 91B, number 1

PHYSICS LETTERS

24 March 1980

PION ELECTROMAGNETIC POLARIZABILIT1ES AND QUARKS E. LLANTA and R. TARRACH Departament de F[sica Tebrica, Universitat de Barcelona, Spain Received 18 December 1979

The electric and magnetic polarizabilitiesof the neutral and charged pion are calculated in a coloured quark field theory at the one-loop level. The theory has as free parameter the quark mass but our results do not depend on it. We have found that the electric polarizabilitiesare a~r+-= -0.04 o~/m3, %r0 = -0.4 a/rn 3. These values are compared with calculations in other models and some comments are made about the polarizability sum rules.

It is well known that the Compton amplitude, up to second order in the photon energy, is determined by two parameters, c~ and fl, the so-called electric and magnetic polarizabilities (apart from mass, charge and magnetic moment) (see refs [1,2] for some reviews on electromagnetic polarizabilities). For charged pions almost the only possibility o f measuring these structure constants is from the shift o f energy levels in pionic atoms [1]. It is, however, a rather difficult task because o f the presence of some other reasons for shifting the levels (strong interaction o f the pion with the nucleus, finite dimension o f the nucleus, nuclear polarizability, etc.). For neutral pions the siutation is even worse. There is some hope in extracting some information on the polarizabilities from p h o t o n - p h o t o n scattering near the two-pion threshold in e+e - collisions. There are several values o f aT+ and ~ 0 calculated within different models as we can see in table 1 [8]. There is no good agreement between these values. In the case o f the charged pion the value varies more than one order o f magnitude. In the case o f the neutral one even the sign is not determined. We can also note that the calculations in refs. [7,8] which are made with quarks lead to results lower than the others. The aim of this note is to calculate the electromagnetic polarizabilities o f the neutral and charged pion in a coloured Gell-Mann-Zweig quark field theory at the one-loop level. The soft current PCAC approximation allows one to compute c~ and/3 as functions o f the 132

Table 1

Terent'ev [3] Pervushin and Volkov [4] Gal'perin and Kalinovsky [5] L'vov and Petrun'kin [6] Degtev [7] Efimov and Okhlopkova[8]

a~r+ (odm 3)

e%o (aim 3)

0.16 0.31 0.3 0.25 +- 0.11 0.1 0.014

0 -0.04 -0.06 0.055 -+0.11 -0.07

quark mass, but, in fact, our results will not depend on it. Our results are thus parameter free. As is well known, the g0 _+ 23' amplitude is very well described within an analogous framework. In this case, however, the computation gives an anomaly which is well known to be an exact result in perturbation theory. But also for non-anomalous observables like the charge radius o f the pion and the neutral kaon, a similar calculation gives a result in good agreement with recent experiments [9]. The computation o f the polarizabilities requires the knowledge of the Compton amplitude [10]. The four gauge-invariant tensors for Compton scattering are L~ v = k .

k'

guy

_

kUk,V ,

L~ v = k ' k ' QUQV _ Q . X ( Q U k , U + kUQV) + ( Q ' K ) 2 g uv ,

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PHYSICS LETTERS

24 March 1980

L~ v = k ' k ' (QUkV + k'UQ v)

k/

- Q . K (kUk v + k'Uk'V), L~ v

:

k'Uk v ,

(1) / ~'q

where Q = ½(q + q ' ) , K = ½(k + k ' ) and q and k are the incoming pion and photon four-momenta and the primed quantities correspond to outgoing particles. The expression for the tensor is

l

+ 5 p~rmutato-~

q~

X.~............ (+

4

r uv = - e 2 ~ B i ( k ' k ' , Q ' K ) L ~ v . i=1

1

crOSSed

(2)

,~

The polarizabilities are defined in terms of the amplitudes B 1 and B 2 which go with the first two tensors, which are the only physical ones, according to ref. [11]:

b

ha.

-= (1/2M) (e2/4~r) (B 1 + Q2B2)C , 13 - - - ( 1 / 2 M ) (e2 /4rr)Bel ,

(3)

where the superscript means the limiting value (at k . k' = (2" K = 0) o f the continuum contributions and M is the particle's mass. Due to the structure of L~ v and due to a kinematical zero o f B 4 neither o f the last amplitudes contribute in this limit. It is therefore sufficient to calculate the Compton scattering tensor in the forward direction and then take the limit O . K -+ 0 to obtain a and/3. At the one-loop level the diagrams that will contribute to the n +- Compton scattering are shown in fig. 1. At first sight it seems that diagrams b, c and d are already contained in diagram a because pions are nothing but bound states of one quark and one antiquark. But one should note that a perturbative calculation never gives bound states and therefore we must introduce diagrams b, c and d by hand. As a p r o o f o f this we will see that the result of diagram a for n +- is not gauge invariant, but if we add a, b, c and d, the result will be gauge invariant. In our calculation, PCAC is imposed as a definition of the interpolating pion field. This leads, in the soft current approximation, to an effective coupling between pions and quarks igjkqjjsqk w i t h ( q u = u, qd

-=a)

guu/m = gdd/m = gud/m = - v ~ / f z r m r r ,

(4)

/

//

,-,r

\x

d

~,

/

Fig. 1. Diagrams that contribute to the n +-Compton scattering at the one-loop level Wavy lines are photons, dotted ones are pions and continous ones are quarks. where m is the quark mass and f~ the pion coupling decay constant. As we have noted above, it is sufficient to calculate in the forward direction (k' = k, q' = q) and then take the limit Q . K -+ 0. Furthermore, a and fi only depend on B 1 and B2, the coefficients o f L ~ v and L~ v, which in the forward direction are: L ~ v = _ka k v, L~ v = - ( q " k) (qUkV + qVkU) + (q" k)Zg uu .

(5)

So, B 1 is the coefficient o f - k U k v and B 2 is the coefficient o f - ( q - k) (qUkV + qVkU) and the knowledge of these coefficients is sufficient to calculate a and 13. Thus we will forget about the terms proportional to guy. 133

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PHYSICS LETTERS

The contribution of diagram a is then

zation: GR(P) = G(P) -- G(P)Ip2=mg

e2 [ 2 q U q V _ s kUk v Ta~lr-+ 2 2~.2 f~ m~ uv _

-~E(p)/~PZlp2=

+ 7q" k (qUk v + qVkU)] .

(6)

90m 2 For the positive pion, the electromagnetic vertex is given by

d4p t-lTr X i J(2n)413f

2(p2-m2).

(11)

m r

With this we obtain the contribution of diagram d which, in the soft current limit, is given by: uv - -2 Tdrr-+

1 2 2 2qU qV +'2kUkV +

j-~ mTrTr

5m 2

(qgkV +qVkU

(12)

12em 2 2 2 f~ mr

AU(q, q') -

24 March 1980

J5 ~ - ~ ] 5 / ¢ + ~ ¢ , _ m

/~ + i f - m

2Tr[] u i 3 ig-~-

m

]5

i . i ]} ~S-m lS)ff- i f - m ,(7)

where e < 0 and q and q' are the pion momenta. This integral is divergent and therefore one has to go through charge renormalization. In order to do this we must subtract the coefficient of (q + q')U for q2 = q,2 = m 2, (q, _ q)2 = 0. So A~t(q, q') = A u (q, q ') _ (q + q')U X A q+q ' (q2 = q,2 __ m r2 '

(q, _ q)2

= 0).

(8)

After calculating A~ (q, q') and working in the soft current limit, the contribution of diagrams b and c is easily obtained:

Adding the contributions of the four diagrams we note that there is no contribution of qUqV as is required by gauge invariance. With the above definitions of a and /3, we obtain 1

aTr-+ -

(a +

e2

36 4n

1

1

a

3 f27r2 mTr

0.003-~, m~ 2 a _ 7 e2 1 l~mr) = 0.0008 - 5 ' 45 4~r mTr 3 .-2 2[-2rn! j~Tr mr (13)

where a = e2/47r, and m = 330 MeV. In the case of the neutral pion only diagram a contributes. Its contribution is: T~r° - .,2 e2 2 2 I~-~

(qUkV "~'q V k U ) - s kUkV1 ,(14)

J n m r'i7

and therefore v

_

--e 2

[4q.qU

5 e2

J Trm~r

%r°-

5m 2 (q ukv + qVkU

+q'~

)1

(9)

.

1 i 3 f27r 2 1 8 4 n m~r

a 0.03 m n3 '

e2 1 1 ( m n ] 2 =0.0026 ( a + / 3 ) r O = 5 art m 3 f2rr2 \2m ]

mn

The pion self-energy is given by •

i

-12m2/'~2 2 J . . . Nd41 ~ Trl]5~-s-~Jsg'-zfiml f ~rmn

~zTr)

(lO) where p is the pion momentum. This integral is quadratically divergent but becomes finite after taking into account the mass and the wave function renormali134

First of all, it is of interest to note that an_+ and a~ro are given in the lowest order in m 2 / 4 m 2 and that (a +/3)~r_+ and (a +/3),o start at the next order. Our values for a + t3 are therefore small, which is consistent with the common assumption a +/3 = 0 [ 3 - 5 , 8 ] . They can be compared with the results of ref. [6] : (a +/3)~_+ = (0.35 + 0.05) × 10 -4 fm 3 and (a +/3)7ro = (0.99 + 0.05) × 10 -4 fm 3. We find

Volume 91B, number 1

PHYSICS LETTERS

(a +/3)n_+ = 0 . 1 6 X 10 - 4 fm 3 , (c~ + ~),ro = 0.6 X 10 . 4 fm 3 .

a+fl=-~-I !o 2rr 2

ffrro = - 0 . 4 e~/m 3 .

dco ~ Ot(co),

(18)

(16)

Our values for a, on the other hand, are mass independent. They do not depend on any free parameter. In order to compare them with the ones given in table 1 we have to multiply them by 4rr. This leads to /~+ = - 0 . 0 4 a / m 3 ,

24 March 1980

(17)

For the charged pion we find a sign opposite to all the other results shown in table 1, and a small absolute value. Precisely because the absolute value is small the sign does not mean very much. We conclude that the electric polarizability of the charged pion is very small. For the neutral pion we also find a negative sign, in agreement with most of the other results, but with a much larger absolute value. We conclude that the neutral pion electric polarizability is negative and much larger than previously expected. We would like to stress here that all the other computations of the electric polarizabilities depend on some (or several) more or less unknown parameters. Terent'ev [3] relates the polarizabilities with % a parameter measured in the rr -+ ev'), decay. This is very badly known [12], so much so that one cannot give a reliable prediction of the electromagnetic polarizabilities based on it. Pervushin and Volkov [4] work with pions and baryons instead of with quarks. Furthermore, their polarizabilities are strongly energy dependent. The same happens in Gal'perin and Kalinovsky's approach [5]. Their electromagnetic polarizabilities vary so much with the energy that it is difficult to extract from these approaches an unambiguous value. L'vov and Petrun'kin [6] depend on a huge number of different channels and therefore parameters and furthermore on an assumption of helicity conservation. We are closest to Efimov and Okhlopkova's results [8] which are based on quarks but nevertheless depend very strongly on an essentially unknown parameter. There are theoretical predictions of the electromagnetic polarizabilities based on dispersion relations leading to sum rules which allow a phenomenological determination of a and/3 if certain cross sections are known for an adequate range of energies. There is one polarizability sum rule which gives ct +/3 as a function of the total photoproduction cross section [ 13]. It is the forward real physical photon sum rule

where co is the photon energy in the laboratory system, coO is the photoproduction threshold energy and ot(co ) is the total photoproduction cross section. This sum rule can be saturated in a reliable way with experimental data for protons but in the pion case at(co ) is not well known. There is also a sum rule based on a dispersion relation for a virtual longitudinal photon Compton amplitude if one assumes that no subtraction is required [14]. This sum rule reads for scalar particles ~

o ~ ( c o , q2)

a=-f dco lim 2rr2 o a 0 q2+0

_q2

1

'

(19)

where %(co, q2) is the total longitudinal photoproduction cross section for virtual photons of mass q2. Notice that scattering experiments determine %(co, q2) for q2 < 0, so the right-hand side is positive and thus a > 0. We have obtained c~ < 0 and therefore this sum rule turns out to need a subtraction, which makes it useless for determining the electromagnetic polarizabilities. One can arrive at the same conclusion working in QED at the lowest nontrivial order [15]. We have performed the first parameter-independent calculation of the pion polarizabilities. It is based on coloured Gell-Mann-Zweig quarks. The charged pion electric polarizability turns out to be very small. The neutral pion electric polarizability on the contrary is ten times larger and negative. One of us (E. L1.) is indebted to the Ministerio de Educaci6n y Ciencia for a grant. References

[ 1] T. Ericson, Interaction studies in nuclei, eds. H. Jochim and B. Ziegler (North-Holland, 1975). [2] P.S. Baranov and L.V. Fil'kov, Sov. J. Part. Nucl. 7 (1976) 42. [3] M.A. Terent'ev, Sov. J. Nucl. Phys. 16(1972) 87. [4] V.N. Pervushin and M.K. Volkov, Phys. Lett. 55B (1975) 405. [5] A.S. Gal'perin and Y.L. Kalinovsky, JINR, P2-10849, Dubna (1977). [6] A.L L'vov and V.A. Petrun'kin, preprint N170, P.N. Lebedev Phys. Inst. Moscow (1977). 135

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[7] V.A. Petrun'kin, Electromagnetic interactions of nuclei at low and medium energy (Nauka, 1976) p. 282. [8] E.V. Efimov and V.A. Ohhlopkova, JINR, E4-11568, Dubna (1978). [9] R. Tarrach, Z. Phys. C (1979), to be published. [10] J. Bernab6u and R. Tarrach, Ann. Phys. 102 (1976) 323. [11] J. Bernab6u, T. Ericson and C. Ferro-Fontan, Phys. Lett. 49B (1974) 381. [12] P. Depommier, J. Heintze, C. Rubbia and V. Soergel,

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Phys. Lett. 7 (1963) 285; A. Stetz et al., Phys. Rev. Lett. 33 (1974) 1455; Nucl. Phys. B138 (1978) 285; D.A. Ortendahl, LBL-5305 Ph.D. Thesis (1976); P. Pascual and R. Tarrach, Nucl. Phys. B146 (1978) 509. [13] M. Damashek and F.J. Gilman, Phys. Rev. D1 (1970) 1319. [14] J. Bernab6u and R. Tarrach, Phys. Lett. 55B (1975) 183. [15] E. Llanta and R. Tarrach, Phys. Lett. 78B (1978) 586.