Three-gluon exchange contribution to forward high-energy scattering

Volume 83B, number 1

PHYSICS LETTERS

23 April 1979

THREE-GLUON EXCHANGE CONTRIBUTION TO FORWARD HIGH-ENERGY SCATTERING M. F U K U G I T A Rutherford Laboratory, Chilton, Didcot, Oxon 0X11 OQX, England and J. KWlECII~ISKI Institute o f Nuclear Physics, 31-342 Krak6w, Poland Received 13 December 1978

The odd charge conjugation part of the three-gluon exchange is shown to give a real amplitude linearly rising with energy in high-energy scattering. This implies necessity of subtraction in the dispersion relation for the crossing odd amplitude. The three-gluon exchange is estimated to give about 3% in the real-to-imaginary ratio for forward pp scattering at ISR energies.

The fundamental assumption o f quantum chromodynamics is the existence o f the gluonic degrees o f freedom, which should play an important role in the high-energy exchange mechanism as well as in the spectroscopy of hadrons. In the vacuum channel one expects two families o f states or exchange corresponding to gluonic and qq degrees o f freedom [1 ] with possible mixing between them. The bare pomeron singularity (C = + 1) is identified with the two- and moregluon exchange contributions [ 2 - 4 ] , while the q~ contribution leads to the reggeon singularity [3]. This contrasts with the dual topological approach [5], where the bare pomeron singularity arises from the renormalization o f reggeons. In the odd charge conjugation channels, we also expect the multi-gluon exchanges as well as the q~l exchanges. Thus a n e w singularity should appear corresponding to the odd charge conjugation part of gluon exchanges (it originates from three- or more-gluon exchanges), in addition to the conventional exchanges which are identified with the reggeons w, ¢ etc. with canonical intercept a <~ ~. The purpose of this note is to discuss the three-gluon exchange contribution to high-energy forward scattering. The three-gluon exchange contains b o t h even and odd charge conjugation components. We consider only the odd-C part. It gives at high energies a purely real

amplitude linearly rising with energy. Higher-order corrections would change this behaviour * 1, presumably leading to some new odd-C Regge singularity with intercept near one. Nevertheless, we may hope that the three-gluon exchange alone gives a rough estimate of this new singularity, like in the case of the two-gluon exchange which has turned out to give a reasonable model [ 2 - 4 ] for the even-C part. The presence o f a real amplitude contribution, linearly rising with energy, invalidates unsubtracted dispersion relations for odd-C (i.e., crossing odd) amplitude • 2 This is to be tested in the forward dispersion #l

.2

Rather little is known about structures of the Regge singular. ities in the odd-C (colour-singlet) vacuum channel, while there have been some studies of the high-energies behaviour for the even-C channel in non-abelian gauge theories with the Higgs mechanism or with a massive field [6]. If the new odd-C contribution gets renormalized ilato the Regge singularity with intercept close to one (a = 1 - e, e > 0), the subtraction is unnecessary. In this case we see, however, that even for negligible contribution O (e) to the imaginary part, which may not be detectable (if e < 1), one obtains a finite 0 (1) real part which effectively appears as the subtraction term. This may readily be seen in the Regge form: A (s) = ieIe-(br/2)a]cos (7rc~/2)]sa~.

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relation for pp and KN scattering. (In KN scattering the new odd-C contribution would also imply strong energy dependence of the K L - K S regeneration phase). In the case of KN scattering, however, one expects suppression of the three-gluon amplitude, since the SU(3) singlet three-gluon system with C = - 1 decoupies from the KK, system in the SU(3) limit, and it would couple only through symmetry breaking. Furthermore, we have also found in the simple eikonal model [4] that the three-gluon contribution still vanishes, for the forward direction, even in the case of broken SU(3) symmetry. So in what follows we shall consider forward pp scattering. The three-gluon exchange contribution to the forward high-energy scattering amplitude may most easily be derived in the infinite momentum formulation of the field theory, which leads to the eikonal formalism [7,4]. The amplitude for proton-proton scattering is constructed in terms of the quark-quark amplitudes given by one-gluon exchange and the proton wave function in the infinite momentum frame. The eikonal operator ~ is a sum of quark-quark "potentials", 2

{a,]l,]Ii} (/I) (/II) l/(r/I -- r/II)'

(1)

with the two-dimensional transverse coordinates of quarks in proton I/II rli, i I = +}b - x3I, II{°I, II

+ (x21,II/(XlI,tt + x2I,II)) P t,II,

23 April 1979

[41. The two-gluon contribution and the odd-C part of the three-gluon exchange, however, can be obtained from the eikonal expansion, as if the longitudinal integrations were done beforehand (see eqs. (1) and (3)). The forward pp scattering amplitude A(s, t = 0) is given by

A (s, t = O) =

-2isf d2b A (b),

(4)

where the partial-wave amplitude A (b) is the appropriate average of the eikonal operator over the infinite momentum proton wave function: A ( b ) = ( 2 ! ) 2 f [ I dxiidxii I l

XS(1-Exii)(l-

EXiil)

(5)

X d2pld2plld26old2~Oli(~l~li lexp(i2) l ~i~ii ) ,

with ~ the normalized wave function:

t~ = 6-1/2eabc ~ ( ( x i } , p, o3).

(6)

The three-gluon system couples to quarks in a proton in three different ways as illustrated in figs. 1a 1c. The vertex corresponding to the coupling of all three gluons to the same quark (fig. 1a) is momentum independent. The coupling in fig. lb is proportional to the "form factor" F(k3):

r2i,l I = +-½b -- x3i, iiwi,ii -

-

(XII,II/(XII,I I +X2I,II))PI,II,

r3i,i I = + }b + (Xli,i I + x21,ii)toi,ii •

(2)

Here b is the impact parameter, ~l,II and PI,II the relative transverse coordinates, and xii,11 the longitudinal momentum fractions of quark i. We take the lagrangian of the form 22 = g~U(ka/2) ~A a. The potential V is the Fourier transform of the gluon propagator

k 1 k2 k 3

k~ k7 k 3

(o)

(b)

G(k), v(x)

-

1 (~r)2

f d2keik-x G(k)

(3)

In non-abelian theory care is needed in the treatment of the ordering in the expansion of the eikonal formula 120

k T k2 k 3

Fig. 1.

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23 April 1979

linearly with energy in the forward direction. (The sign is negative in ~p scattering.) On the other hand, the two-gluon contribution is given by the following formula: 7A1261 =2

Fig. 2.

is

"2~-'

X fd2kld2k2(2Tr)26 (2) (k 1 +k2)

F(k3) = f d2pd2co eik3"P

Dxi) ,

X IT({x},p, co)l 2 I-I dx i 2!6 (1 -

X [1 - F ( k 2 ) ] 2G(kl)G(k2).

/

i.e., the same factor appeared in the two-gluon exchange [4] ,3 (fig. 2). In order to compute the vertex function in fig. lc we need to specify the w dependence of the wave function. We compute it under the following assumptions: (i) Non-relativistic approximation, i.e., we put xi=~ 1/3 wherever it is necessary. (ii) The wave function ~0 depends on a single variable s = (r 1 - r2)2 + (r 2 - r3) 2 + (r 3 - rl)2. With these assumptions the vertex function G(kl, k2, k3) corresponding to fig. 1c is related to the form factor F as

We assume a simple pole formula for the form factor F following ref. [4], F(k 2) =A2/(A 2 + k2),

i=l

After some manipulations the final expression for the three-gluon contribution to forward pp scattering reads:

s

1

5

\47r2! "~.. "6 3

3

X fi=lI-I d2ki(27r)28(2)(i~ 1= ki) 3 X [1-~F(k2)+2F(l~k2)l i=1

(9)

2

3 "I-IG(ki). i=1

We note here that the three-gluon exchange contribution

(C = - 1) in pp scattering is real and positive and increases ,3 Notice that our form factor defined by eq. (7) is different from f u s e d in ref. [4] by a factor 2 in the argument, i.e.,

F(k) = [(2k).

(I I)

with A = mo/2 (this corresponds to the 0-meson pole dominance of the electromagnetic form factor). We observe that the integrand for eq. (9) vanishes as k i 0, while that for eq. (10) remains finite in this limit. Since eqs. (9)and (10)are infrared-divergence free, the gluon mass is set equal to zero, and the bare gluon propagator is given by

G(k) = 1/k 2 .

I__A[3G]=2(g21

(10)

(7)

(12)

Taking gluon self-energy and vertex corrections into account, we then replace the coupling constant g2 with the running coupling constant ~2, g2 (k2) =~2(/~2)/[1 + 9(gZ(la2)/167rZ)ln(k2/p2)] (because of ignorance of the infrared behaviour, we simply put ~(k 2) = ~-(gt2) for k2< /~2). These corrections are important in A [3G], where the main contribution to the integral comes from the relatively large k 2 region, while they are rather small in A [2G]. The coupling constant is estimated from the pp total cross section Otot(pp) ~ A [2G//is, where A [2G] is written from eq. (10) as A [2GI = A-2 ((~2)2/27r)~-

(13)

~"being the correction factor (~"= I, ify(k 2) is set constant). For Otot(pp ) = 27-40 mb ,4 and p = 0.5-0.4 GeV we obtain

, 4 If A [ 2G] is to be identified with the bare Pomeron contribution, the energy-independent part o f the total cross section (e.g., a(pp) ~ 27 mb in ref. [8]) should be used.

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~(/j2)2/47r = 0 . 7 - 0 . 9 ,

[A[3GI/A [2G] 1= 3%--5%.

(14)

(If the corrections are not taken into account, i.e., if a ( k 2) is set constant, then the ratio A [3G]/A [2G] increases by about a factor three.) Since A [2G] is to give the total cross section, the ratio A [3G]/A [2G] should give the contribution to the real-to-imaginary ratio ppp at very high energies:

PP

A~

°tot

'

where (I/s) [A[2G] [ is the energy-independent part of OtPoP t. In the dispersion-relation calculation of the real part O[3pG] is to be ascribed to the subtraction term in the crossing odd amplitude. For Otot = 27 mb (see footnote 4) and for t2 = 0.5 GeV we obtain p[3G] ,~ 2.7% pp

(16)

in the ISR region. Our typical value, 3%, obtained for p[3G], is significant, but of the order of the experimental error of ppp measured in the ISR region [8]. The presence of the subtraction term implied by three-gluon exchange would be tested in a precise dispersion-relation analysis with high-precision data. The knowledge of the real part of p~ scattering at very high energy will allow us to test directly the presence of this new odd-C real term.

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23 April 1979

We thank all our colleagues for discussions. We are also grateful to D.E. Soper for his useful correspondence. One of us (J.K.) thanks Roger Phillips and Chan Hong Mo for their hospitality at the theory division of the Rutherford Laboratory, and another (M.F.) thanks Zbigniew Bochnacki and Wieslaw Czyz for their hospitality at the Institute of Nuclear Physics in Krack6w.

References [1 ] T. Appelquist and H.D. Politzer, Phys. Rev. Lett. 34 (1975) 43; R.L. Jaffe and K. Johnson, Phys. Lett. 60B (1976) 201 ; P. Hasenfratz and J. Kuti, Phys. Rep. 40C (1978) 77 and references therein. [2] F.E. Low, Phys. Rev. D12 (1975) 163. [3] S. Nussinov, Phys. Rev. D14 (1976) 246. [4] J.F. Gunion and D.E. Soper, Phys. Rev. DI5 (1977) 2617. [5] G.F. Chew and C. Rosenzweig, Phys. Rep. 41C (1978) 265; Chan H.M. and Tsou S.T., Rutherford Lab. preprint RL78-080 (1976)and references therein. [6] M.T. Grisaru, H.J. Schnitzer and H.-S. Tsao, Phys. Rev. Lett 30 (1973) 811; H.T. Nieh and Y.-P. Yao, Phys. Rev. Lett. 32 (1974) 1074; B.M. McCoy and T.T. Wu, Phys. Rev. Lett. 35 (1974) 604; L.N. Lipatov, Yadernaya Fiz. 23 (1976) 642; J. Bartels, Phys. Lett. 68B (1977) 258; J.B. Bronzan and R.L. Sugar, Phys. Rev. D17 (1978) 585. [7] J.D. Bjorken, J.B. Kogut and D.E. Soper, Phys. Rev. D3 (1971) 1382. [8] U. Amaldi et al., Phys. Lett. 66B (1977) 390.