Simple mechanism of softening structure functions at low transverse momentum region

10 December 1998

Physics Letters B 443 Ž1998. 387–393

Simple mechanism of softening structure functions at low transverse momentum region S.V. Molodtsov

a,1

, A.M. Snigirev

b,2

, G.M. Zinovjev

c,d,3

a

State Research Center Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia b Nuclear Physics Institute, Moscow State UniÕersity, 119899 Moscow, Russia c Fakultat ¨ fur ¨ Physik, UniÕersitat ¨ Bielefeld, 33615 Bielefeld, Germany d BogolyuboÕ Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Ukraine Received 14 May 1998; revised 23 September 1998 Editor: P.V. Landshoff

Abstract The relevance of non-abelian dipole configurations of quarks in forming nucleus structure functions is discussed. It is shown that radiation generated by dipole configurations while moving relativistically along their axes is described by distributions which are finite and infrared stable in a low transverse momentum region. They approach exponentially the perturbative regime of large transverse momenta and the rate of this transition is defined by the distance between the dipole charges in its rest frame. q 1998 Elsevier Science B.V. All rights reserved. PACS: 24.85.q p; 12.38.Mh; 25.75.q r Keywords: Classical bremsstrahlung; Structure functions; Non-abelian dipole

Theory of dense relativistic gluon Žparton. systems has been recently recognized to be crucially important in solving many Ževen long term pending. problems of high energy hadron physics w1x. But on the eve of a new round of experiments with ultrarelativistic nuclear beams Žat RHIC and LHC. this subject becomes a real challenge because of an extreme necessity to reliably answer the question about the initial conditions and early evolution of the quarkgluon plasma Žif it is formed. as emphasized in w2x. Apparently clearly, an answer depends essentially on

1

E-mail: [email protected] E-mail: [email protected] 3 E-mail: [email protected], [email protected] 2

the behaviour of the nuclear structure functions which are calculable perturbatively in QCD w3x. However, extrapolating these results, even to the region of moderate parton transverse momenta, takes very special efforts w4x and leads to singular distributions for the small momentum values. Moreover, a rather provocative idea in this context launched in w5x suggests, that for the largest nuclei at very high energy, the initial parton density could become so high that the intrinsic transverse parton momenta reach magnitudes where an overlap with the region mastered by perturbative QCD already occurs. It treats the structure function of nucleus as an aggregate of quantum fluctuations on the ground of classical gluon fields generated by ultrarelativistic Žanti-.quarks, and argues that the respec-

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 2 9 0 - 8

S.V. MolodtsoÕ et al.r Physics Letters B 443 (1998) 387–393

388

tive classical beam bremsstrahlung could modify many results which are discussed as possible quarkgluon plasma signatures. In the course of such calculations w5,6x, light cone QCD is handled in the light cone gauge w7x because of taking the limit of Õ ™ 1. This becomes quite nontrivial problem w8x and the corresponding potentials turn out pretty singular and have no the direct physical meaning. In the present paper, at first, dealing with classical electrodynamics in order to make the results somewhat easily understandable, we show that the field strengths of particular dipole configurations regularized by averaging Ereg , Hreg justify taking the immediate limit of Õ ™ 1 for physical observables and provide reasonable asymptotic behaviour for the structure functions. Then, the analysis is extended to non-abelian gauge theory with a description of colour bremsstrahlung in full analogy with classical electrodynamics being argued and eventually demonstrated for the corresponding configurations of colour charges. We believe it might be indicative to appraise nonperturbative region role contributing to the behaviour of nuclear structure functions in the McLerran-Venugopalan approach. It is well known that the field of a classical charge freely moving with the constant velocity Õ close to 1 Ž c s 1. is strongly compressed in the motion direction. At Õ s 1 the field strengths disappear Žequal to zero. besides the plane z s x 0 s t Žwhen moving in z-direction. where the transverse components tend to blow up unlike the potential which does exist, though is singular in this plane. Actually, a nonexisting finite limit of physical observable quantity, like the field strengths in this case, signifies simply that a real physical charge can’t be accelerated up to the light velocity because of an infinite amount of energy necessary to do so. Let us turn to a system of randomly distributed charges with the center of mass ultrarelativistically moving in the direction of the unit vector n. Its potential is given by the following integral over the charge density distribution q Ž y . Žin the rest frame of system. demonstrating obviously the cylindrical symmetry of a solution in the Minkowski space, i.e. um Am Ž x . s dy q Ž y . X , Ž 1. R Ž x, y.

H

where um s Ž u 0 , u., u s Õu 0 n, u 0 s Ž1 y Õ 2 .y1 r2 ,

the compressed distance RX Ž x, y . s Žyx 02 q Ž x y yX . 2 q wyx 0 u 0 q Ž x y yX ,u.x 2 .1r2 and the vector yX absorbs the Lorentz contraction of the charge system in the factor uy1 in the motion direction, thus 0 yX s Ž yn. nru 0 q y y Ž yn. n. In the particular case of a dipole with charges e and ye being placed at the points y1 s ydr2, y2 s dr2 of the dipole rest frame, the electric field components become E Hs e u0 E< < s

e X3

R1

½

½

X3

y

x H yd H r2 RX23

R1

u 0 Ž xn . y Õu 0 x 0 q

e y

x H qd H r2

X3

R2

½

Ž dn .

u 0 Ž xn . y Õu 0 x 0 y

2

,

5

Ž dn . 2

5

5

,

Ž 2.

with RXi s RX Ž x, yi ., i s 1,2. Although these fields are singular in the limit Õ ™ 1, nevertheless, one may notice their orthogonal Žrelative to the motion direction. components are developing two symmetric peaks opposite-orientated at < d H < s 0. Let us now trace the reaction of a fast moving particle passing through the field of such a configuration. Due to the Lorentz contraction this particle is exposed to two very short d-like pulses of opposite signs. If the contraction is large enough that the time interval T ŽT ; dru 0 . between two pulses is less than the classical radius of massive particle a s e 2rm there is not enough time to respond to both separate pulses and the charge is sensitive to the smeared Žaveraged in time. field only. Indeed, the evolution of charge radiation is controlled by the Newtonian equation with the radiative friction w˙ s g Ž t . q 2 e 2 wr3m , ¨ and its general retarded solution is w s w0 q

t

`

Hy` g Ž t . dt qHt

g Ž t . e Ž ty t .r k dt ,

if g Ž t . is the impulse generated by an external field, k s 2r3 a. When two d –like contra-peaks of the amplitude ; G separated in time ; T at the condition Trk < 1 are present, the particle velocity w differs from the initial value w 0 by the quantity of G T 2rk 2 order Žin the order of Trk the pulses cancel each other.. Since G ; Õu 0 G 0 and T ; T0rÕu 0

S.V. MolodtsoÕ et al.r Physics Letters B 443 (1998) 387–393

with G 0 and T0 being fixed in the rest frame, then in the limit Õ ™ 1 it occurs that G T 2rk 2 ™ 0. Hence, it is well grounded to write down for the general solution w s w0 q

`

t

Hy`² g Ž t . : dt qHt ² g Ž t . : e

Ž ty t .r k

dt ,

where ² g Žt .: is an impulse smeared in time. Thus, as the regularized field we should treat one averaged over the interval w x 0 i , x 0 f x s wŽ xn.rÕ y Ž dn.r2 Õu 0 , Ž xn.rÕ q Ž dn.r2 Õu 0 x, i.e. defined by x0 f Õu 0 ² E: s E dx 0 . Ž 3. Ž dn . x 0 i

H

In particular, if the charge distribution takes such a form that ² d H : ; 0 Žaveraging over charge density., then we have ² E< < : s

e

Ž dn .

½

1 < xH<

1 y 2 2 Ž dn . q x H

1r2

5

Ž 4.

It is interesting to notice that our procedure of regularizing by averaging is similar, in a sense, to the potential regularization as done in Ref. w9x dealing with the light-cone variables and then widely used w5,6,10x Žsee Ref. w11x for details.. The radiation energy of a fast moving classical charge in an external electromagnetic field is determined as w12x ^´s

2

2 e14 Ž E q z1 = H . y Ž z1 E .

Hy`dt 3 m

2

1 y z12

d De

2 e14

² E < < :2 . Ž 5. dt 3m 2 here we neglect, of course, a particle influence on a dipole in the first approximation. Integrating Eq. Ž5. over the impact parameter we have s

d 2 k H 2 e14

d De s

H Ž 2p .

2

3m 2

E <2< Ž k H . ,

Ž 6.

where

² E H : f 0 ,² H H : f 0 .

`

done in the framework of relativistic dynamics.. Any other terms should be omitted in the solution of a dynamical equation with radiation force included. Let’s trace now the consequences of this averaging procedure. The same factors Ž1 y Õ 12 . available in the numerator and denominator are cancelled out and the energy radiated per unit time comes about to be independent of the particle energy

dt ,

389

E < < Ž k H . s dx H eyi Ž k H x H . ² E < < :

H

4p e s d kH

½ (

where e1 and z1 are the charge and velocity of the radiating particle, respectively. Here we are interested in the situation when a particle and an ultrarelativistic dipole oriented along its motion direction are going towards each other. In accordance with previous consideration, we should substitute the average field instead of instant values of field strengths in the radiation formula. Such replacement results in the disappearance of the singular part of radiation from the transverse components of field, because ² E H : f 0, ² H H : f 0 and we must take into consideration small longitudinal radiation ‘surviving’. This substitution arises as a leading term of perturbative expansion in Trk which we are able to argue based on the Newtonian equation Žthough it can be

2

p

Ž d kH.

1r2

5

K 1r2 Ž d k H . ,

Ž 7. and K 1r2 Ž z . is the modified Bessel-function. Then asymptotic behaviours of the field are the following lim E < < Ž k H . ™ 4p e ,

k H™0

2

,

1y

lim E < < Ž k H . ™

k H™`

4p e d kH

 1 y eyd k H 4 .

It results in the finite value of the distribution of the radiated photons as obtained from Eq. Ž6. divided by the photon energy k 0 , and being proportional to the Fourier component of electric field squared, of course, in the limit k H ™ 0. It exponentially ap2 proaches the perturbative like behaviour ; 1rk H at the transverse momenta while large enough. Let’s emphasize the rate of transition to the asymptotic regime is regulated by the distance between the dipole charges in its rest frame and the radiation energy is proportional to Ž e 6 .. The x distribution Ž x is a portion of longitudinal momentum carrying by an individual parton. takes well-known ‘soft’ form dk 3rk 0 f dxrx, because 2 d Ž z y x0 . ´ k0 s k3 4 k H . In order to get beyond

390

S.V. MolodtsoÕ et al.r Physics Letters B 443 (1998) 387–393

this approximation we need to keep the next order terms of the Ž1 y Õ 2 .-expansion. Moreover, in lieu of the coherent summation of the fields as in Eq. Ž4. one should average them taking into account the phase shifts between different points, however, these delicate problems require a fully detailed investigation in the future. Actually, the analysis performed may be extended to non-abelian theory. We start on the Yang-Mills equation with external sources Žtheir solutions have been studied in Refs. w13x. &

D m Gmn s j˜n ,

Ž 8.

where j˜m is the density of the external-source current and G˜mn s Em A˜n y En A˜m q gA˜m = A˜n is the strength tensor of the triplet A˜m of gluon fields Žwith the vector product in ‘isotopic’ space in the case of SUŽ2. gauge group. where the covariant derivative is defined as D mw˜ s E mw˜ q gA˜m = w˜ . At small relative velocities of the sources the gluostatic approximation in O ŽŽ Õrel . 0 . order is correct w14x and means for the 4-current of non-abelian dipole configuration j˜0 s r˜ s P˜1 d Ž x y x 1 . q P˜2 d Ž x y x 2 . , where P˜i , x i , i s 1,2 are the ‘ vectors’ of external charges and their coordinates. These ‘ vectors’ are pretty suitable to define convenient natural basis, P˜1 , P˜2 , P˜3 s P˜1 = P˜2 on which the space of equation solutions is spanned. The zeroth component A˜0 s w˜ of the gluon field then appears as a linear combination of the particle ‘charges’

w˜ Ž x ,t . s w 1 Ž x . P˜1 Ž t . q w 2 Ž x . P˜2 Ž t . ,

Ž 9.

Ž t ' x 0 . whereas the vector field is proportional to the third component of the basis vectors A˜Ž x ,t . s a Ž x . P˜3 Ž t . .

Ž 10 .

The potentials and vector field introduced are now the functions of spatial coordinates and two parameters fixing where the particles are stuck up: w i Ž x < x 1 , x 2 ., aŽ x < x 1 , x 2 .. Indeed, the condition

E m j˜m q gA˜m = j˜m s 0 ,

Ž 11 .

providing a self-consistency of the Eqs. Ž8. results in, generally speaking, the basis rotation in the ‘isotopic’ space around the vector V˜ s w ) 1 P˜1 q w ) 2 P˜2

with the frequency < V˜ <. The coefficients w ) 1 s w 1Ž x 2 . and w ) 2 s w 2 Ž x 1 . in the latter are defined by the potential values at the points where the charges reside. The system of gluostatics equations factorized looks Žin terms of the functions above introduced a, w T s 5 w 1 , w 2 5. as follows DDF s d ,

= = = = a s gF JDF ,

Ž 12 .

where the column F is the difference between the columns w and w ) Ž w ) T s 5 w ) 1 , w ) 2 5., i.e. F s w y w ) . Besides, here D k l s =d k l q gaCk l Ž k,l s 1,2. is the covariant derivative, d T s 5 d Ž x y x 1 ., d Ž x y x 2 .5, and C, J are the following 2 = 2 matrices Cs

y P˜1 P˜2

ž / ž P˜ P˜ / 1

1

y P˜2 P˜2

ž / ž P˜ P˜ / 1

2

,

Js

0 y1

1 , 0

where the parentheses stand for scalar products of vector-charges in ‘isotopic’ space and d means the delta-like charge sources of the unit intensity. One can easily see from Eq. Ž11. that C˙ s 0 Žthe dot over C means a time derivative here and for any quantity met in what follows as well., hence, the factorized system of equations is really non-contradictory. In fact, the system should be supplemented with boundary conditions. The delta sources of the righthand sides of Ž12. are eliminated by the corresponding choice of the field w as a superposition with the Coulomb solution wc , i.e. w X s wc q w . An experience gained with calculating and analysing the gluostatic equations in the trivial vacuum w14x reveals that at the large coupling constant g we should approximate a point-like source with more accuracy, in particular, taking into account an induced charge density w15x. However, at the coupling constant small enough the above-mentioned superposition of Coulomb solution and simple boundary conditions in the points of the charge locations comes about fully relevant to reach rather fast convergence of iterations and reasonably high accuracy of the calculations. The quantities w ) were treated as parameters in the course of these calculations. An advantageous feature of the rotating basis is that the boundary conditions at spatial infinity Žwhen the calculation is performed in a large box. are specified in very simple form as w < G ™ 0, ar , z < G ™ 0. The results of the quantitative analysis of the Coulomb-like solutions

S.V. MolodtsoÕ et al.r Physics Letters B 443 (1998) 387–393

may be found in Ref. w14x. When the coupling constant Ž g 2rŽ4p . - 1. is not very large the solutions immovable in group space basis are pretty similar to classical electrodynamics. The major term is given by the Coulomb solution w˜ s F 1Ž x . P˜1 q F 2 Ž x . P˜2 , and hereafter the three basic vectors P˜i , i s 1,2,3 are already time-independent after the global UŽ1. gauge transformation. Thus, we may rewrite as the result what obtained above

½

E˜ H s u 0 P˜1 E˜< < s

P˜1 RX13

½

q

P˜2

x H qd H r2 X3

R1

q P˜2

u 0 Ž xn . y Õu 0 x 0 q

X3

R2

½

x H yd H r2 X3

R2

Ž dn . 2

u 0 Ž xn . y Õu 0 x 0 y

gs

ž P˜ P˜ / q ag P˜ 1

2 3

2

4p r

6 y p 2r2 16p

; 0.02 ,

,

as

g2 4p

&

&

D 2 bm yDm Dn bn y 2 gG˜mn = b˜n s 0 .

Ž 15 .

The gauge fixing suitable here is as follows

Em b˜ m y gA˜m = b˜ m s 0 .

Ž 16 .

Then the vector component of Eq. Ž15. becomes yb¨˜ q ^ b˜ y 2 g w˜ = b˜˙ y g 2w˜ = w˜ = b˜ s j˜,

Ž 17 .

where the current is induced by the particle motion and takes the form

,

5 5

.

Ž 13 .

The corrections of O Ž g 3 . come from gluomagnetic field a which behaves, as a whole, like a field of a constant magnet with the poles residing at the points of charge locations. This component might be estimated, for example, at the interaction potential to be as w14x Vint s

tion of the Ž Õrelrc .1 order. The linearized version of the Yang-Mills equations has the form

j˜s P˜1z1 d Ž x y x 1 . q P˜2 z2 d Ž x y x 2 . .

Ž dn . 2

5

391

,

Ž 14 .

where r s < r <, r s x 1 y x 2 . Such a form of the solution corresponds to one of possible gauge fixing which, apparently, looks more relevant to describing physics just we consider above. Making emphasis further correspondence with electrodynamics let us turn to the problem of radiating non-abelian charges. We studied the dipole gluon field in gluostatic approximation with small relative velocity Õrelrc ™ 0. According to the electrodynamical scheme, the next terms in the Õrelrc perturbative expansion should describe a radiation. Let us rewrite the gluon field as A˜Xm s A˜m q b˜m , with A˜m for gluostatic field and b˜m for small correc-

It obeys the self-consistency condition due to F ) i s 0, i s 1,2. Restricting ourselves to sufficiently small values of coupling constant, we may neglect the O Ž g < w˜ <. terms comparing to b and justify wellknown electrodynamical answer for a radiation. In fact, the solution of linear system Ž17. without the terms of additional rotation taken into account has the retarded form < xyy< 1 dy b˜ Ž x ,t . s j˜ y,t y . Ž 18 . c < xyy< c At the distances R s < x < rather far from the charges we have, as usual, < x y y < s R y yn q . . . , where n s xrR, and the phase factor, therefore, is ct y R q yn. The solution Ž18., thus, becomes 1 1 yn d b˜ Ž x ,t . s dy j˜Ž y,t . q dy j˜Ž y,t . , Rc Rc c dt Ž 19 .

ž

H

/

H

H

where t s t y Rrc. Keeping, naturally, the nonvanishing terms at the infinity only, we come to the following prescription for every charge j˜™ EtE r r˜ . In particular, the dipole contribution is 1 d b˜ Ž x ,t . s dy y r˜ Ž y,t . Rc dt 1 s P˜1 x˙ 1 q P˜2 x˙ 2 . Ž 20 . Rc Then the simple transformations lead to the following result for the radiation intensity of two particles

H

ž

m2 dI s

4p c

3

ž

P˜1 m1

/

q

P˜2 m2

2

/

2 Ž r¨ = n . d V ,

Ž 21 .

392

S.V. MolodtsoÕ et al.r Physics Letters B 443 (1998) 387–393

where m s m1 m 2rŽ m1 q m 2 .. Clearly, Eq. Ž21. is in full correspondence with the electrodynamical results for the bremsstrahlung of a charge while moving along a finite trajectory, and justifies the claim about the specific character of gluostatic solutions. Nevertheless, there is the essential distinction with electrodynamics where the particles of the same charge-tomass ratio do not radiate in the dipole approximation. Clearly, this is not the case of gluodynamics except the special configuration P˜1 s yP˜2 . In principle, now we are able to recalculate the results of what concerns the radiation of fast moving charged particles in the field of non-abelian dipole. However, we already have enough qualitative arguments to conclude the result is just the same for the configurations close to the abelian theory Žwhen all the particles have almost the same orientation in the ‘isotopic’ space. in O Ž g .. We would like to remember here that the electrodynamical radiation results simply from an acceleration of moving particles, unlike the radiation provoked by the colour rotation Žthat is of the O Ž g 3 . being like the gluomagnetic contributions to the interaction potential Ž14. which are ignored in ‘hit’ colour mode studies considered in KR of Ref. w6x and in Ref. w16x together with any analysis the instabilities.. The perturbative parameter fixing the ‘order of non-abeliancy’, as shown, is just the angle between the ‘isovectors’ of corresponding charges, and one should essentially reproduce the electrodynamical results with the collinear ‘isovector’ charges. Thus, using the classical electrodynamics as an example we have demonstrated above the limit of Õ ™ 1 does exist for the physical observable quantities, which are the average field strengths of dipole configurations oriented along their propagation direction Ž‘longitudinal’ dipole., and this qualitative picture does not change when non-abelian theory is considered. In a sense it justifies some intuitive representations about the structure of classical gluon fields generated by ultrarelativistic nuclei collision, and about nuclear structure functions developed by McLerran and Venugopalan in w5x. The gluon transverse momentum distribution here obtained agrees qualitatively with the distribution of Ref. w10x, though essential differences are that we obtain the infrared stable and finite result at k H ™ 0 and the transition to the perturbative like behaviour occurs at the trans-

2 verse momenta k H ; dy2 ; mp2 . We are free to take the minimal size of hadron as the mean distance between dipole charges in its rest frame. Unfortunately, we are unable now to point out the well grounded mechanism to orientate the colour dipoles along their propagation, but we believe Žand may argue. the rather plausible hypothesis is that the dipoles are aligned along the external accelerating field, because such configurations in electrodynamics are energetically more favourable. Let us now summarize our main results. We suggest the regularization procedure of classical fields as being generated in ultrarelativistic heavy ion collisions and relate the field smearing to the Lorentz contraction of field configurations. If characteristic field configuration ‘size’ is less than the classical particle radius, then this particle is able to ‘feel’ an average field only. Operating with the physical observables we have immediately calculated the average field of fast moving ‘longitudinal’ dipole ŽEq. Ž4.. in the limit Õ ™ 1 and then the radiation energy of a fast moving charge in this field ŽEqs. Ž6. and Ž7... The transverse momentum distribution of the photons radiated happens to be finite at small values and exponentially runs into the asymptotic regime 2 2 at larger k H . Applying proper quark-anti; 1rk H quark configurations for the calculation of structure functions leads to the same result as for the photons distribution and agrees qualitatively with result obtained earlier in Ref. w10x. Unlike the light-cone method, every stage of our approach is physically meaningful because we are dealing with observable values only. Actually, including next order terms of expansion in Ž1 y Õ 2 ., could illuminate the origin of structure function behaviour at small x, this we see as the subject of our further investigation.

Acknowledgements This work has been initiated by numerous discussions of classical non-abelian theory and the McLerran-Venugopalan model with V. Goloviznin, M. Gyulassy, I. Khriplovich, Yu. Kovchegov, A. Kovner, A. Leonidov, A. Makhlin, L. McLerran, J. Qiu, D. Rischke, H. Satz and R. Venugopalan. The financial support of RFFI ŽGrants 96-02-16303, 9602-00088G, 97-02-17491. and INTAS ŽGrant 930283, 96-0678. is greatly acknowledged.

S.V. MolodtsoÕ et al.r Physics Letters B 443 (1998) 387–393

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