New threshold effects in hadron-hadron collisions at high energies

Physics Letters B 275 (1992) 491-494 North-Holland

PHYSICS LETTERS B

New threshold effects in hadron-hadron collisions at high energies A.V. A n i s o v i c h a n d V.V. A n i s o v i c h st. PetersburgNuclear PhysicsInstitute, Gatchina, SU-188 350St. Petersburg, USSR Received 12 September 1991; revised manuscript received 11 November 1991

We discuss threshold effects in the p!0 scattering amplitude due to the production of new superheavy particles. It is shown that the production of new particles should be accompanied by screening effects that change significantly the scattering amplitude.

This leads to difficulties in the simultaneous explanation of measured values of ato, andp=ReA/lmA for energies x/~= 0.05-2.0 TeV in the framework of a new threshold hypothesis.

The large value of the real part of the forward pp scattering amplitude measured by the UA4 Collaboration at x / s = 546 GeV, p=0.24_+ 0.04 [ 1 ], has initiated the discussion about the existence of a new threshold at high energies [ 2 - 4 ] : this threshold can be related to the production of some new superheavy particles. This problem continues to be actual up to now [ 5 ]. Besides, at superhigh energies a new strong interaction physics may start [6] so it is important to foresee possible effects in a,o, and p caused by a new threshold. The new particle production process provides an schannel singularity in the scattering amplitude. However, the behaviour of the amplitude near the singularity would be limited by the unitarity condition. The unitarity condition constraints are especially important at superhigh energies. The high energy scattering data analysed in terms of the impact parameter representation give evidence of such a type of interaction in which the scattering amplitude has a maximal inelasticity at the impact parameters b<~ bo(s), where bo(s) increases as ln"s with ½~
count screening due to the s-channel unitarity. Different versions of threshold behaviour are analysed based on the realistic p~) scattering amplitude for the energy region x/~=0.05-2.0 TeV. The scattering amplitude in the impact parameter representation is defined as A (q, s) = 2 I d2b e x p ( - i q . b ) f(b, s ) ,

f(b, s) --i{1 - t/(b, s) exp [2i6 (b, s ) ] ) .

(1)

Total, elastic and inelastic cross sections are expressed in terms o f ~ and q as follows: atot = 2

fd2b

( 1 - q cos 2d) ,

aej= f d2b ( 1 - 2 q c o s 2d+t/2) , ainel= f d2b (1-~/2) .

(2)

We analyse the scattering amplitude in the K-matrix type representation extracting directly the elastic channel:

f(b,s)=

2~(b,s)K(b,s) l-i@(b,s)K(b,s)

"

(3)

Here q~ is the pp phase space factor while the function K contains inelastic contributions only (see fig. 1a). Because of the inelastic channels K is a complex function. We introduce q~ in eq. (3) in order to stress

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metrical scaling model, see for example refs. [ 12-14 ]. We have performed also the calculations for the case with a change of sign ofp at low energies, namely with b g Fig. 1. Blocks which correspond to the production of "old hadtons'" (b) and new heavy particles (c). the way the pl0 state is extracted from K. Redefining the K-function, it is possible, of course, to use q~= 1. The a m p l i t u d e f ( b , s) in the form ofeq, (3) satisfies the s-channel unitarity, and the parameters q and can be written as functions of Re K and Im K. The maximal inelasticity (~/--,0) corresponds to K = i th Z with Z>> 1. In QCD-motivated phenomenology the saturation o f f ( b , s) at bO). In order to investigate new particle production processes it is convenient to extract them from the Kfunction directly: let us replace in eq. (3) K(b, s) -~K(b,s) +a(b,s) (see figs. l b a n d lc) whereK(b,s) corresponds now to the production of the "old hadrons" only (fig. lb), and a(b, s) is related to the production o f new particles (fig. lc). So

f(b, s) = 2

K(b,s)+a(b,s) 1-i[K(b,s)+a(b,s)]

(4)

Let us specify our model. For the function K(b, s) we use a type of parametrization which provides us with a reasonable description of the data in the case

~0(s)=0.9i+0.1(1

2 2 5 G' e V s 2)

this choice of (0(s) gives the same results in the energy region x / s > 50 GeV. The used parametrization of r2(s) and ~0(s) gives a sufficiently good description of the elastic diffractive cross section as well: for example, we have a ' = 0.20 GeV-2 and B = 17 G e V - 2 at ~Js= 1.8 TeV in agreement with the data [ 15]. Because of the saturation o f f ( b , s) at small b, the amplitudef(b, s) is sensitive to a(b, s) at rather large b. The large values of b in the energy region not far from threshold can be caused by the diffractive production mechanism of new particles. To be specific we suppose this mechanism in our evaluations of new threshold effects. The block a(b, s) as a function o f s has a threshold singularity at s--so. We use in our calculations the splane threshold singularity of the type (s-so) 2 × In ( s - s o ) which corresponds to a three-particle intermediate state and should be considered as the strongest one for diffractive production processes (a stronger s-plane singularity gives a larger bump inp). The root singularity ~ corresponds to a twoparticle state in the s-wave and works in the point b = 0 only where the screening is maximal. We neglect its contribution. We parametrize a(b) in the following form:

a(b,s)=O: K(b)=

~o(s)exp[-b2/rZ(s)] 2 +i~0(s) exp [ -b2/rZ(s) ] "

a(b) =

R s)

,

(6)

(5)

It results in a scattering amplitude of the form

A (q, s) = 2nr 2(s)~o(s) exp [ - ¼r2 (s)q 2 ] . Present-day data do not contradict the idea of maximal growth of hadron total cross sections [Im A (0, s) ~ In2s and p - 1/In s]. However, this is not the case in the energy region considered here, , , ~ ~ 0.05-2.0 TeV, At these energies a,o, increases as In s while a decrease of p is not seen. In accordance with these facts we use r 2 (s) = ( 16.5 + 38.9/x/s + 3.16 In ~ x / ~ ) GeV -2 and ~0(s) = 0.9i+ 0.1; it leads to a type ofgeo492

c,,a,,(s) exp ,,=,

where c,, are constants, the functions a,,(s) have the threshold singularity at S=So, and the b2-dependence is supposed to be exponential. The threshold bump in p depends on R z: the larger the value of R 2, the bigger the bump in p. We accept the diffractive mechanism for new particle production and put R 2 = ~r2(s). In the region of x / s ~ 0 . 6 1.0 TeV it gives the values R2~5 GeV -2 which coincides with the slope of the diffractive production o f "old hadrons" at moderate energies. The functions a,,(s) are chosen in the form

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al(s)=(l_l)Z(1 2

x+l

)

~ln ~

30 January 1992

1

+i - - / ' , ( x ) ,

2 70

az(s)=75

1-

1( 1-- 1)( - - -+1l n i I/ x -1- l ;l ) ----Tf1P3 ( x )

II

- - -

lit

......

./

/"

~v

-~lnlx-ll+i--1p2(x),

.2"~,~*'1

, , ~7:::J--

2

a3(s) = ~

,

(7)

~6°50 t:)

where x=s/so and

P,(x)=2x-'-4

40

,

P_, ( x ) = x

3- ~ x3 -2+

P3(x).=X

5 ---3X-4.

lx-'+l~, i

F ~x 1 -3

+ ~ x1

-2

+ ~ x1

-I

+~o .

Eqs. (7) give a,(s) at S>So while one should omit the imaginary parts of the right-hand sides of eqs. (7) for the case s> so whereas the functions a2 (s) and a3 (s) give the possibility to change the production cross section near threshold. Figs. 2 and 3 show the results of our calculation of ~r,o, and p for the following sets of (Cl, c> c3): '

'

,,i

'

'

,

,

/i--:

0.25

'

,

,

'f

l

m ......

T

/: ,r,,

/,,

0.20

',

//'

"-..

0.15

I ~

;'/"

Q 0.10

_.-o.~

i

/ ".,. /

0.05 t

i

LIt

100

J

i

i

i

r

t

t l l

i

1000

v/S(GeV)

Fig. 2. Ratio p = Re A / I m A for the cases I-IV. The data are from refs. [ 15,16].

f

i

i

[

i

100

i

i

i

,/-g(GeV)

i

i

i

LI

i

1000

Fig. 3. pp tolal cross sections for the cases I - I V . The data are

from refs. [ 15,16].

case I = (0.27, 0, 27),

case I I I = (0.2, 2, 0)

case I l = (0.22, 5.5, 0),

caseIV=(0.3,0,7.5).

(8) We put ~ 0 = 500 GeV and R 2 = ~r2(s) for the cases, I, II and IV. In the case III we use R 2 = lr2(s). The variants 1-III are the examples when the maximal value o f p near x / s = 5 5 0 GeV is close to 0.22. The calculated total cross sections for these cases are larger than the experimental ones in the region x / s = 0 . 5 - 1 . 0 TeV. In variant IV we show the case when the calculated values of Cqo, are near the error bars of the experimental data, however, in this case p = 0 . 1 9 8 at , / s = 550 GeV. These results differ from the calculation of new threshold effects of ref. [ 17 ] where screening corrections were not taken into account. The cross section of the new particle production, a~n~, is proportional to Im a and can be extracted from ame~. It is not small in the region x / s ~ 0 . 6 - 1 . 0 TeV for all cases considered: ain,e'~~ 7 mb for the cases l - I l l and ai~" ~ 5 mb for case IV. This indicates that the production of new particles is accompanied by the decrease of the production of the "old hadrons". In conclusion we summarize. The plb scattering amplitude at the energies , , ~ = 0 . 0 5 - 2 . 0 TeV is calculated in the framework of the new threshold hypothesis. The assumptions have been made to en493

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large t h r e s h o l d effects ( w e h a v e u s e d t h e d i f f r a c t i v e production mechanism near the threshold and a s t r o n g t y p e o f s - p l a n e s i n g u l a r i t y ) . In t h e c a l c u l a t i o n s p e r f o r m e d t h e s - c h a n n e l u n i t a r i t y c o n d i t i o n for t h e a m p l i t u d e s is t a k e n i n t o a c c o u n t . It r e s u l t s in a large s c r e e n i n g effect n e a r t h e n e w t h r e s h o l d w h i c h prev e n t s o n e f r o m a g o o d fit o f a~o, a n d p s i m u l t a n e o u s l y . H o w e v e r , m o r e p r e c i s e d a t a are n e e d e d to c o n c l u d e definitely about the absence of the new threshold in the energy region considered. O u r t h a n k s goes t o A.A. A n s e l m a n d Ya.I. A z i m o v for useful r e m a r k s . O n e o f us ( V . V . A . ) is g r a t e f u l t o E. P r e d a z z i for t h e k i n d h o s p i t a l i t y e x t e n d e d to h i m d u r i n g t h e 4 t h Blois W o r k s h o p w h e r e d i s c u s s i o n s h a d s t i m u l a t e d t h i s work.

References [l] UA4 Collab.. D. Bernard et al., Phys. Len. B 198 (1987) 583. [2] K. Kang and S. Hadjitheodoris, in: Proc. 2nd Intern. Conf. on Elastic and diffractive scattering (Rockfeller University, New York, 1987). [3] P.M. Kluit, in: Proc. 2nd Intern. Conf. on Elastic and diffractive scattering (Rockfeller University, New York, 1987). [4]A. Martin, in: Proc. 2nd Intern. Conf. on Elastic and diffractive scattering (Rockfeller University, New York, 1987).

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[5] M. Haguenauer, UA 4/2 experiment: a new measurement of the p value, talk 4th Blois Workshop on Elastic and diffractive scattering (La Biodola, Isola d'Elba, Italy, May 1991). [ 6 ] S. Hagesawa, Fermilab CDF Seminar, ICR-report no. l 5187-5 (1987), unpublished; G.B. Yodh, in: Elastic and diffractive scattering, Proc. Workshop evanston, IL, 1989), Nucl. Phys. B (Proc. Suppl.) 12 (1990) 277; F. Halzen, report no. MAD/PH/504 (1989), unpublished. [7 ] T. Kanki, K. Kinoshita, H. Sumiyosbi and F. Takagi, Prog. Theor. Phys. (Suppl.)A 97 (1989) 1; A. Donnachie and P.V. Landshoff, Nucl. Phys. B 267 ( 1986 ) 690. [8] L.N. Lipatov. Sov. Phys. JETP (1986) 904. [9] T.K. Gaisser and F. Halzen, Phys. Rev. Lett. 54 (1987) 1754, [ 10 ] T.K. Gaisser and T. Stanev, Phys. Lett. B 219 ( 1989 ) 375. [ 11 ] M.G. Ryskin, Nucl. Phys. B (Suppl.) 12 (1990) 40. [12] J. Dias de Deus, Nucl. Phys. B 59 (1973) 231. [13] U. Amaldi and K.R. Schubert, Nucl. Phys. B 166 (1980) 301. [14]P. Kroll, in: Elastic and diffractive scattering, ed. B. Nicolescu and J. Tran Thanb Van (Editions Fronti6res, Gifsur-Yvene, 1986), p. 145. [15] E710 Collab., N. Amos et al., Phys. Rev. Lett. 63 (1989) 2784; S. Shekhar, Preliminary results on p from Fermilab Experiment 710, talk 4th Blois Workshop on Elastic and diffractive scattering (La Biodola, Isola d'Elba, Italy, May 1991 ). [ 16] UA5 Collab., G.J. Alner et al., Z. Phys. C 32 (1986) 153. [17 ] S. Hadjitheodoridis and K. Kang, Phys. Lett. B. 208 (1988) 135; K. Kang and A.R. White, Phys. Rev. D 42 (1990) 835; preprint ANL-HEP-PR-91-32 ( 1991 ).