Can Gluons Trace Baryon Number ?

Home , Baryon, Gluon

CERN-TH/95-343

BI-TP 95/42

CAN GLUONS TRACE BARYON NUMBER ?

D. KHARZEEV

Theory Division, CERN, CH-1211 Geneva, Switzerland

and

Fakultat fur Physik, Universitat Bielefeld, D-33501 Bielefeld, Germany

Abstract

QCD as a gauge non-Ab elian theory imp oses severe constraints on the

structure of the baryon wave function. We p oint out that, contrary to

a widely accepted b elief, the traces of baryon numb er in a high-energy

pro cess can reside in a non-p erturbative con guration of gluon elds, rather

than in the valence quarks. We argue that this conjecture can b e tested

exp erimentally, since it can lead to substantial baryon asymmetry in the

central rapidity region of ultra-relativistic nucleus-nucleus collisions.

CERN-TH/95-343 BI-TP 95/42

In QCD, quarks carry colour, avour, electric charge and isospin. It seems

only natural to assume that they also trace baryon numb er. However, this latter

assumption is not dictated by the structure of QCD, and therefore do es not need

to b e true. Indeed, the assignment of the baryon number B =1=3 to quarks is

based merely on the naive quark mo del classi cation. But anyphysical hadron

state in QCD should b e represented by a state vector which is gauge-invariant{

the constraint which is ignored in most of the naive quark mo del formulations.

This constraint turns out to b e very severe; in fact, there is only one wayto

construct a gauge-invariant state vector of a baryon from quarks and gluons [1]

(note however that there is a large amount of freedom in cho osing the paths

connecting x to x ):

Z Z

x x

ij k

B = P exp ig A dx q (x ) P exp ig A dx q (x )

1 2

x x

1 2

i j

P exp ig q (x ) A dx : (1)

The \string op erators" in (1) acting on the quark eld q (x ) make it transform

as a quark eld at p oint x instead of at x . The tensor then constructs a lo cal

colour singlet and gauge invariant state out of three quark elds (see Fig.1a).

The B in eq. (1) is a set of gauge invariant op erators representing a baryon

in QCD. With prop erly optimised parameters it is used extensively in the rst

principle computations with lattice Monte Carlo attempting to determine the

nucleon mass. The purp ose of this work is to study its phenomenological impact

on baryon numb er pro duction in the central region of nucleus-nucleus collisions .

It is evident from the structure of (1) that the trace of baryon numb er should

b e asso ciated not with the valence quarks, but with a non-p erturbative con g-

uration of gluon elds lo cated at the p oint x - the \string junction" [1]. This

can b e nicely illustrated in the string picture: let us pull all of the quarks away

from the string junction, whichwekeep xed at p oint x. This will lead toqq

pair pro duction and string break-up, but the baryon will always restore itself

around the string junction. The quark comp osition of this resulting baryon will

in general di er from the comp osition of the initial baryon. It is imp ortantto

note that the assignment of the trace of baryon numb er to gluons is not only a

feature of a particular kind of the string mo del, but is a consequence of the lo cal

gauge invariance principle applied to baryons.

Surprisingly, at rst glance this non-trivial structure of baryons in QCD do es

not seem to reveal itself in hadronic interactions in any appreciable way. This

is b ecause to prob e the internal structure of baryons, we need hard interactions,

which usually act on single quarks. The baryon numb er therefore can always b e

asso ciated with the remaining diquark, without sp ecifying its internal structure.

To really understand what traces the baryon numb er, one needs therefore to 2

study pro cesses in which the ow of baryon numb er can b e separated from the

owofvalence quarks.

In this Letter we suggest that studies of baryon pro duction in the central ra-

pidity region of ultra-relativistic pp and esp ecially AA collisions provide a crucial

p ossibili ty to test the baryon structure. Our ndings may prove imp ortant for

understanding the prop erties of dense QCD matter pro duced in AA collisions at

high energies. In what follows, we shall rst formulate our ideas qualitatively,

and then give some quantitative estimates based on the top ological expansion of

QCD [1, 2 ] and Regge phenomenology.

Let us consider rst an ultra-relativisti c pp collision in its centre-of-mass

frame, which coincides with the lab frame in collider exp eriments. At suciently

high energies, the valence quark distributions will b e Lorentz-contracted to thin

pancakes with the thickness of

z ' ; (2)

x P

where P is the c.m. momentum in the collision, and x 1=3isatypical

fraction of the proton's momentum carried bya valence quark. The typical time

needed for the interaction of valence quarks from di erent protons with each

other during the collision is given by the characteristic interquark distance in the

impact parameter plane, t = const O (1 fm). However, the time available

int

for this interaction in the collision is only t z (x P ) . It is therefore

coll V V

clear that at suciently high energies, when t << t , the valence quarks of

coll int

the colliding protons do not have time to interact during the collision and go

through each other, p opulating the fragmentation regions. In the conventional

picture, the baryon numb er follows the valence quarks.

At rst glance, the argument lo oks correct, and is well supp orted exp erimen-

tally - the leading e ect for baryons in high energy pp collisions is well established.

However, the structure of the gauge-invariant baryon wave function (1) suggests

that this scenario may not b e entirely consistent. As wehave stressed ab ove,

in QCD the trace of the baryon numb er has to b e asso ciated with the non-

p erturbative con guration of the gauge eld. This \string junction" contains an

in nite numb er of gluons which, therefore, by virtue of momentum conservation,

should carry on the average an in nitely small fraction x << x of the proton's

S V

momentum. We therefore exp ect that the \string junction" con guration may

not b e Lorentz-contracted to a thin pancakeeven at asymptotically high energies

(see Fig.2), since

z ' >> z : (3)

S V

x P

In this case the string junction will always have enough time to interact, and we

may exp ect to nd stopp ed baryons in the central rapidity region even in a high- 3

energy collision (of course there will also b e baryon-antibaryon pair pro duction

in addition). This argument leads to a p eculiar picture of a high-energy pp

collision: in some events, one or b oth of the string junctions are stopp ed in the

central rapidity region, whereas the valence quarks are stripp ed-o and pro duce

three-jet events in the fragmentation regions. Immediately after collision, the

central region is then lled by a gluon sea containing one or twotwists, which

will later on b e dressed up by sea quarks and will form baryon(s). Note that

the quark comp osition of the pro duced baryons will in general di er from the

comp osition of colliding protons.

Why then is the leading baryon e ect a gross feature of high-energy pp col-

lisions? The reason may b e the following. The string junction, connected to all

three of the valence quarks, is con ned inside the baryon, whereas pp collisions

b ecome on the average more and more p eripheral at high energies. Therefore,

in a typical high-energy collision, the string junctions of the colliding baryons

pass far away from each other in the impact parameter plane and do not inter-

act. One can however select only central events, triggering on high multiplicity

of the pro duced hadrons. In this case, we exp ect that the string junctions will

interact and may b e stopp ed in the central rapidity region. This should lead to

the baryon asymmetry in the central rapidity region: even at very high energies,

there should b e more baryons than antibaryons there.

Fortunately, the data needed to test this conjecture already exist: the exp er-

imental study of baryon and antibaryon pro duction with trigger on asso ciated

hadron multiplicity has b een already p erformed at ISR, at the highest energy

ever available in pp collisions [3]. This study has revealed that in the central ra-

pidity region, the multiplicities asso ciated with a proton are higher than with an

antiproton by ' 10%. It was also found that the numb er of baryons in the central

rapidity region substantially exceeds the numberofantibaryons [4]. These two

observations combined indicate the existence of an appreciable baryon stopping

in central pp collisions even at very high energies [3].

Where else do we encounter central baryon-baryon collisions? In a high energy

nucleus-nucleus collision, the baryons in each of the colliding nuclei are densely

packed in the impact parameter plane, with an average inter-baryon distance

1=2 1=6

r ' A ; (4)

where is the nuclear density, and A is the atomic numb er. The impact parameter

b in an individual baryon-baryon interaction in the nucleus-nucleus collision is

therefore e ectively cut o by the packing parameter: b r . In the case of a lead

nucleus, for example, r app ears to b e very small: r ' 0:4 fm, and a central lead-

lead collision should therefore b e accompanied by a large number of interactions

among the string junctions. This may lead to substantial baryon stopping even

at RHIC and LHC energies. 4

We shall now pro ceed to more quantitative considerations. In the top ological

expansion scheme [1], the separation of the baryon number ow from the ow

of valence quarks in baryon-(anti)baryon interaction can b e represented through

a t-channel exchange of the quarkless junction-antijunction state with the wave

function given by

Z Z

i j

x x

2 2

0 0 0

J i j k

M = P exp ig A dx P exp ig A dx

ij k

0 0

x x

1 1

i j

P exp ig A dx : (5)

The structure of the wave function (5) is illustrated in Fig.1b - it is a quarkless

closed string con guration comp osed from a junction and an antijunction. In

the top ological expansion scheme, the states (5) lie on a Regge tra jectory; its

intercept can b e related to the baryon and reggeon intercepts [1]:

; (6) (0) ' 2 (0) 1 + 3(1 (0)) '

B R

where the baryon intercept (0) has b een set equal to 0, and the reggeon inter-

cept (0) equal to 1=2.

The M exchange should dominate the proton-antiproton annihilation at high

energies [1]. Indeed, annihilatio n requires the baryon numb er transfer in the

tchannel. Conventionally, this corresp onds to the baryon exchange with the

intercept (0) ' 0. Since in Regge theory the energy dep endence of the cross

(0)1 J J

section is given by s , and (0) > (0), the M exchange should give

0 0

the dominant contribution at high energies (see Fig.3), leading to the following

energy dep endence of the annihilation cross section:

(0)1 1=2

s s

ann

' (7)

s s

0 0

instead of the s dep endence implied by conventional baryon exchange (s '

is the usual parameter of Regge theory). Up to the highest energies where 1 GeV

the annihilation can still b e exp erimentally distinguished from other inelastic

pro cesses, the energy dep endence (7) is con rmed by the data. Moreover, the

entire di erence b etween the total pp andpp cross sections can b e attributed to

annihilatio n (see [5] for a recent review).

Let us now turn to the consideration of baryon stopping in pp collisions. The

relevant diagrams are shown in Fig.4; we consider the simultaneous stopping of

the two string junctions in the central rapidity region, accompanied by three-jet

events in the fragmentation regions (Fig.4a), and the stopping of the junction

of one proton in the soft parton eld of the other, accompanied by one three-jet 5

event (Figs.4b,c). To calculate the cross sections, it is convenienttoevaluate the

discontinuity of the corresp onding three-particle elastic scattering pro cess [6,7]

(see Fig.5). Intro ducing the four-momentum p of the pro duced baryon (or the

total momentum of the pair of baryons in the diagram of Fig.5a) and the four

momenta of the colliding protons p , p wehave the following expressions for the

1 2

invariant energy in the proton-baryon systems:

s ' sm e ;

1 t

s ' sm e ; (8)

2 t

where s =(p +p ) is the c.m.s. energy squared of the pp collision, y is the c.m.s.

1 2

2 2 2

rapidity of the pro duced baryon(s), and m is its transverse mass m = m + p .

t B B

The pro duct of the invariants (8) satis es the relation

s s ' m s: (9)

1 2

Let us denote the coupling of the M reggeon and p omeron to the proton by

M P MM 2 MP 2

G and G resp ectively, and intro duce scalar functions f (m ) and f (m )

p p B t B t

J J J

describing the \two baryons - M M " and \one baryon - M -Pomeron"

0 0 0

vertices. The standard calculation [6, 7] then allows us to calculate the cross

sections; for the diagram of Fig.5a we get

2 (0)2

3 (2)

d sm

M 2 MM 2

E : (10) =8[G (0)] f (m )

p B t

d p s

B 0

Analogous calculation for the sum of diagrams in Figs.5b,c gives

! J

(0)+ (0)2

3 (1)

d sm

M P MP 2

E =8G (0)G (0) f (m )

p p B t

d p s

B 0

J J

exp[y ( (0) (0))] + exp[y ( (0) (0))] : (11)

P P

0 0

Using the value (6) of the M intercept, we nd that the double baryon pro duc-

1=2

tion cross section (10) has s energy dep endence and (within the central

rapidity region, where our considerations apply) do es not dep end on rapidity.

The cross section (11) of single baryon stopping also decreases with energy, but

much more slowly.Writing down the Pomeron intercept as

(0)= 1+; (12)

1=4+=2

one gets the energy dep endence of the cross section (11) in the form s .

Atvery high energies the pro cess of single baryon stopping will therefore b e more

imp ortant. Note however that the ISR data show a large value of the correlation

between the probabilities of the stopping of the b eam baryons, indicating that 6

the pro cess of double baryon stopping may still dominate even at rather high

energies. The rapidity dep endence of (11) do es not show a \central plateau",

indicating instead a \central valley" structure. This is in accord with the ISR

data; moreover the particular rapidity dep endence of (11) repro duces the data

[8] reasonably well.

The functional dep endence of the single baryon stopping cross section sim-

ilar to (11) has b een advo cated b efore [9] in a di erent approach; the authors

considered a sp eci c mechanism of p erturbative destruction of the fast diquark,

accompanied by the baryon number owover a large rapidity gap. In the frame-

work of their approach, the authors of ref.[9]have also p erformed a calculation

of the cross section, based on the combination of p erturbative technique and

constituent quark mo del.

Unfortunately, since we b elieve that the dynamics of the stopping pro cess is

genuinely non-p erturbative, so far wehave not b een able to nd a reliable wayof

computing the constant G (0) entering the expression (11) and can only extract

it from the existing data. However once it is done, we can p erform parameter-free

extrap olation to higher energies.

As wehave already stressed ab ove, the formulae (10,11) refer to the net baryon

numb er, i.e. they refer to the di erence b etween the numb er of pro duced baryons

and antibaryons. The pro cess of baryon-antibaryon pair pro duction will therefore

represent an imp ortant background to baryon stopping; the data [8] show that

at ISR energies the probability of baryon stopping is ab out three times smaller

than the probability of baryon pair pro duction. At high energies, the dominant

contribution to the BB pair pro duction will b e given by the interaction of two

Pomerons, with the cross section

2 (0)2

3 (BB )

d sm

P 2 PP 2

E ; (13) =8[G (0)] f (m )

p t

d p s

B 0

representing an energy-indep endent fraction of the total cross section:

(0)1

BB tot

: (14)

Since the top ological structure of the Pomeron is that of a cylinder [1]

P = Tr P exp ig A dx ; (15)

(1)

at high energies the asso ciated multiplicities in the pro cesses of single (n ) and

(2)

double (n ) baryon stopping will b e higher than in the average inelastic event,

describ ed by the cut of the Pomeron exchange diagram (see Fig.6):

5 3

(1) inel (2) inel

n ' n ; n ' n ; (16)

4 2 7

as we already discussed ab ove at the qualitative level. Also, since the baryon

stopping and baryon pair pro duction arise in our scheme from di erent kinds of

t-channel exchange (M and the Pomeron, resp ectively), we exp ect a small corre-

lation b etween the events with the pair pro duction and stopping. Exp erimentally,

it was found to b e 0:16 0:22 [8].

Before we turn to the discussion of nucleus-nucleus collisions, it is useful to

recall the geometrical picture of high energy scattering in the impact parameter

plane. In the impact parameter representation, the growth of the total cross

section at high energies as describ ed by one-Pomeron exchange can b e attributed

to the increase of the e ective radius of the interaction, according to

R ' 2R + ln(s=s ); (17)

int 0

p P

where R is a constant and is the slop e of the Pomeron tra jectory. (The

Froissart b ound allows even faster growth of the interaction radius with energy:

R ln(s=s ).) The central region of the disk b ecomes completely blackat

int 0

high energy if (0) > 1; a further growth of the interaction strength in the

centre of the disk is prevented by the unitarity constraint imp osed on the partial

amplitudes.

The colliding nuclei, the transverse plane nucleon distributions in which are

characterized by the packing parameter (4) will therefore see each other as uni-

form black disks. This means that in a central nucleus-nucleus collision the cross

section of the inelastic nucleon-nucleon collision will not further increase with

energy when R >> r since the soft p eripheral interactions building up the

int

Pomeron will b e e ectively screened out. On the other hand, the pro cesses of

baryon stopping are central in the impact parameter plane, and therefore may

not b e screened in the case of nuclear collisions. A very slow decrease of the cross

section (11) with energy implies then that even at LHC energies the nuclear

stopping may still b e present, as we shall now discuss.

The ISR data [10, 11 ] show that at s =53 GeV the cross sections of proton

and antiproton pro duction at y = 0 and p =0:6 GeV =c are

3 p 3 p

d d

2 2

(y =0)=0:7000:162 mb GeV ; (y =0)=0:4300:033 mb GeV :

3 3

d p d p

(18)

Since the mechanism of baryon-antibaryon pair pro duction (see (13)) obviously

leads to an equal numb er of protons and antiprotons, the data (18) imply that

the following fraction of the protons in the central rapidity region is pro duced by

stopping:

p p

f ( s =53 GeV )= '25%: (19)

p p

+ 8

We can now estimate the ratio R of multiplicities asso ciated with proton (n ) and

antiproton (n ) pro duction, using the predictions (16). Since the multiplicity

asso ciated with protons and antiprotons should b e the same in the absence of

stopping, we get

(1) p

n n

; (20) R = =(1f )+f

st st

p p

n n

where wehave omitted the contribution of the double stopping pro cess, which

is asymptotically suppressed at high energies according to (10), but may still

b e imp ortant at ISR energies [8]. Assuming that the multiplicity asso ciated

with antiprotons do es not di er substantially from the average multiplicityofan

p inel

inelastic event, n ' n , in accord with (13) and with exp erimental data [3],

we obtain from (20) and (16) an estimate

R ' 1:05: (21)

Even though the value (21) agrees within exp erimental errors with the measured

value of R ' 1:1 [3], wemay conclude that the contribution of double baryon

stopping with higher asso ciated multiplicity (16) is p ossible. A detailed exp er-

imental study of double baryon pro duction in pp collisions would therefore b e

useful to clarify the situation.

Extrap olation to the pp collisions at the energies of s ' 6 TeV (corresp ond-

ing to the c.m.s. energy p er nucleon-nucleon collision in Pb-Pb interactions at

LHC) according to formulae (11, 13) with = 0:08 [12] yields then the following

stopping fraction:

f ( s =6 TeV ) 5%: (22)

The nucleus-nucleus collisions will b e accompanied bymuch larger stopping, as

we discussed ab ove, and we exp ect that the estimate (22) in this case can only b e

considered as a lower b ound on the baryon asymmetry. Therefore wemay exp ect

substantial excess of baryons over antibaryons in the central rapidity region of

nucleus-nucleus collisions at LHC.

It is interesting that already at SPS energy, the otherwise successful phe-

nomenological approaches based on the top ological expansion [13-16], but not

taking into account the presence of the string junction explicitly, seem to under-

estimate [17 ] the pronounced baryon stopping observed exp erimentally in nucleus-

nucleus collisions [18].

It would b e useful to analyse the dynamics of baryon stopping in an approach

where the top ological structure of the baryon is explicit: the Skyrme mo del. The

formation of baryon-antibaryon pairs in this approach is treated as the forma-

tion of top ological defects in the quark condensate [19] (the net baryon number

of the pro duced pairs is of course equal to zero). The picture prop osed ab ove 9

would invokeinto consideration also the high-energy scattering of such top olog-

ical defects. Since the Lorentz b o ost of such con gurations is not trivial, one

may exp ect the o ccurence of the nal states in which one or two Skyrmions are

stopp ed in the central region, and the part of their pion eld is \shaken o " to the

fragmentation region. Suchevents would contribute to the baryon asymmetry in

the central region. We leave the consideration of the baryon stopping dynamics

in top ological mo dels for further studies.

Amazingly, the so-called \Centauro" and \Chiron" events rep orted in cosmic-

rayemulsion exp eriments [20] and interpreted in favour of the existence of dense

quark matter by Bjorken and McLerran [21 ], are characterized by non-vanishing

baryon numb er density.Aswas formulated in ref. [22 ], \the uid in the central

region has no net baryon numb er, so that there would need to b e a sp ontaneous

generation of net baryon densitytomake these ob jects". Indeed, in the scenario

prop osed by Bjorken [22], at ultra-relativisti c energies the valence quarks of the

colliding nuclei pass through each other, leaving b ehind a \little Universe" of zero

net baryon density, which at the moment of its pro duction contains a gluon sea.

In our picture, this sea from the very b eginning is made stormyby the presence

of non-p erturbativetwists - sp eci c con gurations of the gluon eld, which trace

non-zero net baryon numb er. The \little Universe", just like our big one, may

therefore generate substantial baryon asymmetry.

Iamvery indebted to G. Veneziano for many instructive suggestions and

encouragement. It is a pleasure to acknowledge stimulating and enlightening

discussions with J.-P. Blaizot, A. Cap ella, J. Ellis, K.J. Eskola, M. Gazdzicki,

K. Ka jantie, L. McLerran, J.-Y. Ollitrault and H. Satz, whom I also thank for

his interest in this study. This work was supp orted by the German Research

Ministry (BMBW) under contract 06 BI 721. 10

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