Short-distance structure of hadrons in supersymmetric QCD

Nuclear Physics B214 (1983) 317-349 @~North-Holland Publishing Company

S H O R T - D I S T A N C E S T R U C T U R E OF H A D R O N S IN SUPERSYMMETRIC QCD c. KOUNNAS

CERN, Geneva, Switzerland D.A. ROSS

Department o/Physics, University o/Southampton, Southampton, UK Received 3 November 1982

Assuming supersymmetry to be a real symmetry in particle physics, we study the short-distance internal structure of nucleons (structure functions at large Q2). We determine all calculable quantities in the leading logarithm approximation and discuss the N = 1 supersymmetry relations between them. Further relations are obtained by observing that the N = 1 supersymmetric lagrangian becomes N = 2 supersymmetric if the quarks and squarks are (formally) put into the adjoint representation of SU(3) colour. From the calculated elements of the 4 × 4 matrix of the Q 2 evolution kernels and knowing the quark and gluon distributions at low Q2 we are able to determine the quark, squark, gluon, and ghiino distributions for Q 2 above the threshold for the production of supersymmetricparticles. We find that in the asymptotic limit (Q2 ~ 0c) the quarks carry only 2 of the nucleon's baryon number and the remaining ~1 is carried by the squarks. We also find that in this limit 0.32 of the longitudinal nucleon momentum is carried by giuons, 0.08 by gluinos, 0.36 by the quarks and 0.24 by the squarks.

1. Introduction W h a t do n u c l e o n s look like at short distances? This question can only be answered after the particle c o n t e n t of the theory of strong interactions a n d their short-distance interactions are k n o w n . I n Q C D we k n o w that the n u c l e o n structure f u n c t i o n s are described b y q u a r k a n d gluon distributions with well defined Q2 evolutions. If one believes that s u p e r s y m m e t r y [1 ] is a g e n u i n e symmetry, the particle c o n t e n t s of s t a n d a r d Q C D will be enlarged b y the supersymmetric partners of ghions (ghiinos) a n d of quarks (squarks) [2]. The existence of such particles drastically changes the picture of the h a d r o n i c i n t e r n a l structure at short distances. I n a realistic s u p e r s y m m e t r i c model the s u p e r s y m m e t r y m u s t be b r o k e n [3] i n such a way that all successful p h e n o m e n o l o g i c a l predictions of Q C D a n d electroweak interactions are reproduced. W h a t e v e r the m e c h a n i s m for the s u p e r s y m m e t r y b r e a k i n g (explicit [4], s p o n t a n e o u s [2, 3] or d y n a m i c a l [5]) the masses of the scalar quarks 317

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(squarks) must be larger than 17 GeV [6] and the mass of the fermionic gluons (gluinos) must be larger than 1-2 GeV [5,7-12]. Above these limits the hadronic structure will be significantly altered. In this paper we give a quantitative description of the nucleon structure above the threshold for the production of squarks. We derive the quark and squark distributions using the known quark and gluon distributions at low Q2 [13, 14]. At high Q2 these distributions may be very important in future experiments such as p~ ~/~+/~-X for Q 2 (500)2 GeV 2 or even for large-pv experiments. The quark, gluon, gluino and squark distributions at high Q2 can be used to establish the existence of supersymmetric particles and hence the correctness of introducing supersymmetry into standard QCD or grand unified theories. In order to be able to make this kind of analysis we needed to calculate the 4 x 4 matrix for the Q2 evolution kernel (the inverse Mellin transform of the 4 x 4 anomalous dimension matrix) in the leading logarithm approximation. These 16 quantities are related by N = 1 or N = 2 supersymmetry identities [1]. This is an example of the attractive supersymmetric relations between bosons and fermions. In sect. 2 we explain the formalism used in this paper and give the integrodifferential equations for gluon, gluino, quark and squark distributions. In sect. 3 we give the analytic result of the 4 X 4 matrix of the Q2 evolution kernel as well as the anomalous dimension matrix. We also derive the above-mentioned N = 1 and N = 2 supersymmetry relations. Quantities which are not gauge invariant violate the supersymmetry relations because of the introduction into the effective lagrangian of a gauge-fixing term which explicitly breaks supersymmetry. However, we show by explicit calculation that in the light-cone axial gauge even gauge-dependent quantities preserve the supersymmetry relations. In sect. 4 we study the quark and squark valence distributions, determining them as a function of Bjorken x and Q2 (using the known valence quark distribution at low Q2). In sect. 5 we study the gluon, gluino and "sea" quark and squark distributions. As in the case of the valence distributions we use a simple parametrization of the parton distributions at the squark threshold. Finally, sect. 6 is devoted to a general conclusion.

2. SupersymmetricQCD 2.1. PARTICLE CONTENT OF SUPERSYMMETRICQCD AND THEIR INTERACTIONS The existence of a spin one-half supersymmetry generator implies that every particle must have a "super" partner with spin shifted by a half unit [1,2]. Thus the gauge bosons (gluons) have Majorana fermions as partners. These are in the adjoint representation of SU(3) and are called gluinos. Likewise left- and right-handed quarks have complex scalar partners which are called squarks (s-type and t-type

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associated with left- and right-handed quarks, respectively): gluons [G] ~ gluinos [ ~ ], right-handed quarks [qR, ClR] ~ squarks [t, t ÷ ], left-handed quarks [qL, ClL] ~ squarks [s, s+]. The various interactions between gluons, gluinos, quarks, and squarks are described by the following N = 1 supersymmetric, SU(3) (colour) gauge-invariant, lagrangian

+ ~lii~qi + ( D f i i ) + D s i + ( D J i ) + D ~ t i

+ ig~/2(~RS/- TaqLi + ~aLt~-T~qRi - h.c.) - ½g2(s~ T~si - t? T~ti) 2 + mass terms,

(2.1)

where a runs over the eight gluons and i runs over the number of flavours. A gauge-fixing term plus a Faddeev-Popov ghost term must be added to the above lagrangian. G~ is the gluon field strength = a.GS -

+ gfobccgc;,

and D~ is the covariant derivative =

-

igC;

7,

where T a are the generators of SU(3) in a given representation [for gluinos these are the structure constants of the g r o u p ( T a ) i j = i f aij and for quarks and squarks in the fundamental representation they are the Gell-Mann SU(3) matrices]. The presence of mass terms in the lagrangian breaks supersymmetry. The mass spectrum [2-5, 11] depends on the way the supersymmetry is broken. For exact supersymmetry the gluino is massless and the quarks have the same mass as their corresponding squarks. In practice the gluinos can have a Majorana mass [11] and the squarks have much larger masses than their corresponding quarks, because supersymmetry is broken. For sufficiently large momentum scales (Q2 >> m 2) the effects of these masses may be neglected and supersymmetry is effectively unbroken. The lagrangian density (2.1) contains neither auxiliary fields or unphysical degrees of freedom. This is because we have chosen the Wess-Zumino gauge [1] and integrated out all the auxiliary fields. But such a gauge choice still leaves the usual

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SU(3) gauge to be fixed. One way of doing this is to choose covariant gauges 1

2

GF----~ ( O~,G~) ,

(2.2)

with ~FP = ffa( O~D,n)a. However, this particular gauge explicitly breaks the supersymmetry since the supersymmetry partner of the longitudinal gluon O"G~,is not present in the gauge fixing. Such supersymmetry breaking does not affect gauge-invariant quantities such as the anomalous dimensions of gauge-invariant operators, but the naive supersymmetry relations (such as the equality of the wave-function renormalization of members of the same supermultiplet) are in general violated in such a gauge. In order to preserve the supersymmetry relations for all Green functions (gauge invariant or otherwise) we must use the superfield formalism [1] or axial gauges ( n " A = 0, n 2 :~ 0) [15]. In this gauge all supersymmetry relations can be shown to be obeyed. This follows from the fact that the gluon propagator is a projector,

..(oiT( G;(x )G:( x ))lo) = o,

(2.3)

and so the unphysical fields n~G~ always decouple from Green functions. In sect. 3 we will show by explicit calculation that these supersymmetry relations are also preserved in the light-cone axial gauge (n- A = 0, n 2 = 0). This gauge suffers from the problem of extra infrared divergences in some of the Green functions, which have to be treated by a consistent regularization method.

2.2. PARAMETRIZATION OF THE STRUCTURE FUNCTIONS

In the leading logarithm approximation the factorization theorem and the renormalization group equation can be combined into the following equations for the Q 2 evolution of structure functions [ 14,16]:

F~(x, Q 2 ) = xe~,[qi(x, Q2) + qi(x ' Q2) + si(x ' Q 2 ) + Ji(x ' Q2) +ti(x, Q2)+~(x, Q2)],

(2.4)

where i runs over all flavours whose mass threshold is below Q2 and where eq, are the electric charges of flavour i quarks (or antiquarks) and squarks. In the leading logarithm approximation gluons and gluinos do not couple to the electromagnetic or weak currents.

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The Q2 dependence of these parton densities (quarks, squarks, gluons and gluinos) is controlled by the following set of integrodifferential equations:

871" PGG®G+PGx®~k+PGq®E(qi+~Ii) i + eos ® E ( s , +

Q

+ t, +

2~u~ (x, Q2)=a(Q2)[px~®G+Pxx®X+Pxq®~_(q,+i F)it8~r

i

+exs® E ( s i + gi + ti+ ~)] d "x, Q2) a(QZ)[ 1P Q2~Q-~qi( 8~r ~nf qG®a+

Pqx®~+Pqq®qi

+ ns® (si+ Q 2 ) [ ~ n f eso®G+an1_1 O 2- ~d Si(x'o2)= a ( 8~ f psX®X+2±psq ®qi+Pss®Si]

Q2+ti(x, Q2)=a(Q2) 1--~Psa ® ~ 8~r [--~nfP~®G+4nf + ½Psq ® qi q-

P~ ®t,],

(2.5)

with similar equations to the last two for antiquarks and antisquarks. The kernels P~j are functions of x and the parton densities for gluons (G), gluinos (~), quarks or squarks are functions of x and Q2. The convolution ® is defined by

(qi)

(si, ti)

A®B=

a(y)B

,

(2.6)

In the absence of squarks one can see that the quantities q~ - qj or qi - qi have Q2 evolutions which only depend on themselves. This is because these quantities cannot mix with gluons or gluinos since q~ - qj is a flavour non-singlet and q~ - ~, is odd under charge conjugation. This is not the case in the presence of squarks since the squarks have the same quantum numbers as the quarks (except for spin). In this case

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the quantities qt + st - (qj + sj) or qt + st - (77i+ st) have Q2 evolutions which depend on themselves:

2 d

Q - ~ (qi+si-(qj+sj))

a(Q2)[(pqq+pm)®(qt+st_(q:+sj))], 8~r (2.7a)

2

d

(2.7b)

Q --clQ-~(q'+st-(71t+gi))=ct(Q2)[(Pqq+Psq)®(qt+st-(gli+gi))]8rr

[here and henceforth we write s(x, Q2) to be the distribution of s-type plus t-type squarks]. These equations are derived using the relation (see sect. 3)

pqq + pqs=p

(2.8)

+ Pss,

which follows from supersymmetry. We close this section by defining some useful combinations of quark and squark distributions nf

(i) valence quarks:

qv(x, Q2) = E (qi(x, Q2)-qt(x, Q2)); i=1

(2.9a) nf

(it) valence squarks:

Sv(X, Q2) = E (Si(x, Q2)--Si(x, Q2)); i=l (2.9b)

(iii) total quark and antiquark distribution:

n~

qt(x ' O2) = i•= 1 (qi(x, 0 2) + Uli(X, Q / ) ) ; (2.9c)

(iv) total squark and antisquark distribution:

nf

S,(x, Q2) = E (Si(X, a2)q-Si(X, a2)) " i~l (2.9d)

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The Q2 evolution of qv, s v, qt, st, G and X are given by the equations 2 d Q ~-~qv

°t(Q2) ( p q q ® q v + e q s ® S v ) 8'n"

(2.10a)

2 d (Psq ®qv + Pss ® Sv) Q ~-Q-~Sv= ~t(Q2) 87:

(2.10b)

The quantities qv and s v do not mix with gluons and gluinos. The other quantities mix with each other and their Q2evolution is described by the 4 × 4 matrix equation G 2 d

x

G

ks _

qt

8~r

St

PXG

Pxx

Pxq

Pxs

PqG

Pqx

Pqq

eqs

PsG

Psx

Psq

Pss

®

X (2.11) qt St

In the next section we give analytic expressions for the 16 elements of the kernel matrix and discuss supersymmetry relations between them. 3. Relations and sum rules in supersymmetric Q C D

3.1. ANALYTIC EXPRESSIONS FOR Pij(x) AND SUPERSYMMETRICRELATIONS Our results for Pij(x) are shown in table 1. They are obtained from the inverse Mellin transforms of the anomalous dimensions of twist 2 operators in the Wilson operator product expansion on the light cone. Pi/are gauge-invariant quantities, so that there are some supersymmetry relations between different kernels. Since the lagrangian is N = 1 supersymmetric (i.e. gluons and gluinos are members of the same vector supermultiplet and quarks and squarks are members of the same matter supermultiplet) the quantities

I,~(X, Q 2 ) = G ( X , Q2) .jr_X(x, Q2),

(3.1a)

qt(x ' Q2) +st(x ' Q2)

(3.1b)

~b(x, Q 2 ) =

are invariant under supersymmetry transformations and this leads to the tion equations Q2 ~Q._svd a ( 2)8tr Q [Pw ® v + ev+ ® ~],

Q_~q~=d2

a(Q2)[

Q2 evolu(3.2a)

,

(3.2b)

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The analytic expression of the

r l+x 2 P°°=2CA[~I---;5+ +

~qo= 2 ~ [ x ,

+ (l-

TABLE 1 kernels of eqs. (2.10) and (2. I 1)

Pij(x)

l+(1-x)2-(x2+(1-x)Z)]+[3Cm] x

-

TR]¢$(I

-

x)

~/]

PsG:2TR[I--[X2+(I--x)2]]

[l+V-x/]

PGX= 2CA

X

[ l+x 2 ]

PXX = 2CA[ ~

J + (3CA -- T R ) ~ ( 1 - X)

Pqx = 2TR[1 -- X ] P~x = 2TR[X]

Poq=2CF[ l + (1-- x)2 ] Phq = 2Cv(I - x)

Pqq=2C F[ [ l +~x 2J

]

+2CF~(I-x)

Psq = 2CF[X]

PGs = 2CF

[l+(1-X) X

2

] X

PX~ = 2CF[1] Pqs = 2CF[1]

Pss = 2 c F

[ ~--l+x2 ;3-+ (l-x)]+2CeS(l-x)

The N = 1 and N = 2 supersymmetry relations [eqs. (3.4) and (3.5)] are automatically valid. The same is true for the relations (3.5). The group factors CA, T R and C F for the SU(N) colour with n~t right-handed and n L left-handed flavours in the fundamental representation are given as CA = N, T R = ~(nf~R + nL), CF = (N 2 -

1)/2N.

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325

where Pvv, Pv,, P,v, P,+ are the inverse Mellin transforms of elements of the anomalous dimension matrix of the two supersymmetric invariant operators O~ and ON+, where

O~v=._~.iN [G~a~,aD~,2 . . . D~N ,G/~'N + %a'Y"'D t~2 iN OON=-~. [£1iY"~D" . . . .

. . .

D~'~M] ,

DUNqi + s+D "' " " D"Ns i + tf~D "' "'" D"Nti],

(3.3a)

(3.3b)

where the above operators must be symmetrized in the indices/a 1. . . . . /~u and all traces must be subtracted; D" is the covariant derivative. Combining eqs. (2.11) and (3.2) we have the relations P~ = PC,G + ex6 = PGX + PAX,

(3.4a)

Pv• = PGq + P•q = eGs + gas,

(3.4b)

P+v = PqG + PsG =

(3.4c)

Pqh + Ps+~,

P+¢ = Pqq + Psq = Pqs + ess"

(3.4d)

It can be seen from table 1 that these relations are satisfied. The first relation, eq. (3.4a), was already known from standard QCD (without gluinos or squarks) when the quarks are formally put into the adjoint representations of SU(3), so that the standard QCD lagrangian looks like a supersymmetric lagrangian with one vector supermultiplet. The relations (3.4) are valid up to all orders in perturbation theory provided one uses a renormalization prescription which preserves supersymmetry (e.g. the dimensional reduction scheme, DR, instead of dimensional renormalization scheme MS). We also observe that if C F = CA = TR we have further relations PGi + Phi + Pqi + Psi = PGj + Phj + Pqj + Psj,

(3.5a)

for i, j = G, ~, q, s and

P,j = Pk,,

(3.5b)

where i and k o r j and I are related by the interchange of ~ and q. These relations do not follow from N = 1 supersymmetry but from the fact that if the quarks transform as the adjoint representation of SU(3) (Majorana spinors) then the lagrangian is symmetric under the interchange ~ ~, q and moreover it is invariant under N = 2 supersymmetry transformations where all the fields transform as members of a

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326

single N = 2 s u p e r s y m m e t r y vector multiplet, v 2. The relation (3.5a) n o w follows f r o m the fact that the combination l)2 = G + ~k + qt + s t

is an N = 2 supersymmetric invariant quantity (provided implies that

2 d

Q --d-~ ( G + ~ + q t + s t ) =

a(Q2)P 8~ v2v2® ( G + ~ + q

CF-----CA=TR). +s),

This

(3.6)

where Pv2v2 is the inverse Mellin transform of the N = 2 supersymmetric operator ON__V2-O~ + ON*.

(3.7)

Relation (3.5a) n o w follows from eq. (3.6) and the relation (3.5b) follows f r o m the s y m m e t r y of the N = 2 supersymmetric lagrangian under the interchange X ~ q. If i, j = q or s, then relation (3.5a) holds even without the condition C F = CA = TR; however, this is an accident of leading order perturbation theory. It follows because the group theory factor for all Pq~ or Psi is CF, SO we can relax the condition C F = CA = T R. But in higher orders the kernels will depend on the other independent eigenvalues of the Casimir operators and so this relation will not be valid. In any case relations (3.5) are unphysical but they are useful for reducing the n u m b e r of independent kernels Pij. Since Pi9 are gauge-invariant quantities, relations (3.4) and (3.5) are valid in any gauge. Supersymmetry suggests that the a n o m a l o u s dimensions of fields which are m e m b e r s of the same supermultiplet must be the same: Yv -= Yx = Y~,

(3.8a)

Y~, - "/q = 3's-

(3.8b)

However, these are not gauge-invariant quantities and in covariant gauges, for which the gauge fixing and F a d d e e v - P o p o v ghost terms explicitly break the s u p e r s y m m e t r y invariance, they are not valid. Nevertheless we find that in the light-cone axial gauge these relations are indeed valid:

-2yo=

3Ca -

4cA/lr~O

dx

x -l-t)

3CA - T R - 4CA,

TR - 4C a f01 (1 - xd )x ~-~

3CA - T R - 4CA,

_ 2 y x = 3Ca _ TR _

(1 -

(3.9a)

e

E

(3.9b)

C. Kounnas, D . A . Ross /

- 27q = 2CF-- 4CF fo (1 - -dx x ) l-~

- 27~ = 2C F - 4CF

327

Hadrons in supersymmetric Q C D

2C F

4C F ----e---

(3.9c)

2 C F - 4C---E,

(1 -- x) 1-~

(3.9d)

so that the relations (3.8) are satisfied. Moreover, if the quarks and squarks are in the adjoint representation of SU(3) we have the further relation from N = 2 supersymmetry (3.10)

")tG = "~X = "~q = ~ s I C A = T a - - C F "

From eq. (3.9) we can see that these anomalous dimensions are infrared divergent and we regularize the infrared divergences using dimensional regularization. These divergences always cancel for gauge-invariant quantities, such as the anomalous dimensions of gauge-invariant operators, which are the sum of the pole part of an amputated Green function yr,N and a field anomalous dimension - 2 7 : N = Yro N - 27G, Yc~

(3.1 la)

7~x = 3'~ - 2Yx,

(3.1 lb)

~ ' q ~--- y rq N-

2yq,

(3.11c)

7sN--7 N -rs--

2y s •

(3.1 ld)

Thus, for example,

f01d x x N-1l ( l _ x ) [

7rN=2cA

1 +

[

X 2

'-~

=

E

,]

+2C A -2s(N)+N(N+I)

(3.12) so we see that for Y~x this infrared divergence cancels and one obtains a finite gauge-invariant anomalous dimension

I

~,~x= 2CA - 2 s ( N ) q

1]

N ( ~ ; + 1) + 3CA - TR,

where

N 1 S ( N ) = ~_, -:-. i-1

l

(3.13)

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T h e reason why the naive s u p e r s y m m e t r y relations are valid in the light-cone axial gauge (n.Ga = 0, n 2 = 0) is that the unphysical degrees of freedom n. G a, which explicitly b r e a k supersymmetry, do not p r o p a g a t e so that they always decouple from all G r e e n functions.

3.2. LONGITUDINAL MOMENTUM CONSERVATION

Pij(x) represents

the probability to generate a p a r t o n i from a parton j with a function of longitudinal m o m e n t u m x. In leading order this h a p p e n s by the emission of a p a r t o n k whose longitudinal m o m e n t u m fraction is 1 - x, so in this order we have

Pij(x) --- Pkj(1

- x).

(3.14)

In higher order these relations b e c o m e m u c h more complicated and consequently lose their usefulness. F r o m eq. (3.14) we have

PeG(x) -- Pc,~ (1 = e

- x),

(3.15a)

(3.15b)

o(1 - x ) ,

PqG(X) = PqG(1 - x ) ,

(3.15C)

Psc;(x) = Psc~(1 - x ) ,

(3.15d)

P o x ( X ) = exx(1 - x ) ,

(3.15e)

Pqx(X) = Psx(1 - x ) ,

(3.15f)

PGq(X) = eqq(1 - x ) ,

(3.15g)

e h q ( X ) ~'~ esq(l -- x ) ,

(3.15h)

P~(x) -- P~(1

- x),

(3.15i)

e h s ( X ) = Pqs(1 - x ) .

(3.15j)

3.3. MOMENTUM SUM RULE The total fraction of longitudinal m o m e n t u m carried by all the constituents of a hadron must sum to one:

foldxx[G(x, Q2)+)t(x, Q2)+qt(x, Q2)+st(x, Q2)] =

1.

(3.16a)

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329

Taking the derivative of this with respect to Q2 we have

fodXX[PGi"[-I

PAiq-Pqi-~-Psi]=O,

i= G,X,q,s.

(3.16b)

This relation expresses the fact that the anomalous dimension of the energymomentum tensor is zero (i.e. energy and momentum are conserved). 3.4. BARYON NUMBER CONSERVATION

The fact that the quarks and squarks carry baryon number ½ and that baryon number is conserved implies that ~f01d X ni~1( q i

+ Si --

(for nucleons).

qi - si) = 1,

(3.17a)

Taking the derivative of this with respect to Q2 gives us foldx(Pqq

"q- Psq)=foldx(Pqs

q- Pss) = 0.

(3.17b)

This is the statement that baryonic current is conserved so that its anomalous dimension must vanish. Here we observe a fundamental difference between this supersymmetric model and standard QCD. In standard QCD only the quarks carry baryon number, whereas in the supersymmetric version the squarks can also carry baryon number so that only a fraction of the baryon number is now carried by the quarks. It will be shown in the next section that in the asymptotic limit (Q2 ~ ~ ) the quarks carry 2 of the total baryon number and the squarks carry ½. 4. Valence parton distributions

Having discussed the properties of the Q2 evolution kernels we now turn to the discussion of how the constituents are distributed inside hadrons. We begin with the valence distributions qv(X, Q2) and Sv(X, Q2). These quantities do not mix with gluons or gluinos and their QZ evolution is given by eq. (2.10). We denote by (A>N the Nth moment of A(x): 1 N(A)N = f d x x 1A(X). (4.1) a0 Taking the first and second moments of eq. (2.10) we obtain the following sets of differential equations:

2d Q ~-~

(Sv) '

o/ 2, CF

m

8~r

-1 2

(4.2) --2

1

C. Kounnas, D.A. Ross / Hadrons in supersymmetric QCD

330

2 d ((qv)2]=a(Q

o

<,v)2:

2 )CF

8.

(

3

23

1)(qv2

-4

(Sv}2

(4.3)

The general solution of eq. (4.2) is

(qv(Q2)),=2+((qv(Q2)),-2)exp[-3CFS(Q2, Q2)],

(4.4a)

(sv(Q2))l=l-((qv(Q2)}l-2)exp[-3CFS(QE, Q2)],

(4.4b)

where we have used the fact that the total number of valence partons in a number is 3 and S(Q2, Q2)= ~0 l o g ( l +

a(Q2______~) / 4¢r bolog Q---~-2 Q2]'

(4.5a)

with b0 = 3CA - TR

(= 3 for CA - 3 and TR= n f= 6)*.

(4.5b)

The solution to eq. (4.3) is (qv(Q2))2 = (qv(Q2))2 [3exp[-

3CFS(Q2, Q2)] + 2exp[_~CvS(Q2, Q2)]]

+ (sv(Q2))2 [3exp [ -

3CvS(O 2, 002)] - 3exp[- ~CFS (0 2, 002)]], (4.6a)

(s~(Q2))2 = (qv(002))2 [-35exp[ - 3CFS(Q2, Q02)] _ 3exp[_ ~ CFS (Q2, O 2)1]

+ (Sv(002))2[ 2exp[ - 3CFS(0 2, 002)] + 3exp[ -

~CFS(O 2, 002)]] • (4.6b)

Asymptotically (as Q2..., oo) the quarks carry 2 of the baryon number and the squarks ½. Furthermore the fraction of momentum carried by the valence partons goes to zero; this means that asymptotically all the momentum is transferred to the gluons, gluinos and "sea" quarks and squarks (and antiquarks and antisquarks). This means that asymptotically the valence parton distributions become a delta function at x = 0:

q~(x, a2) Q2___~ ~ ~ 28(x),

(4.7a)

S v ( X , Q2)QE--~ooS(X ) .

(4.7b)

• We define T~ to be ½× (no. of left-handed flavours + no. of right-handed flavours).

C. Kounnas,D.A. Ross / Hadronsin supersymmetricQCD

331

For the large moments (N ~ oo) we have

Q2 d

(

N]

Nj

a(Q2 ) ( - 4 C F l o g N )

8~

(4.8)

Nj'

which implies that the behaviour of the distributions near x = 1 is given by

qv(x, QZ)x_~lqv(X ' QZ)[logx] .f l l'CFS(&,Q~),

(4.9a)

Sv(X, Q2)x--~ Sv(X, Q2)[logl]4CFS(Q2'Q~)"

(4.9b)

If the threshold for the productions of squarks in deep inelastic scattering is Q 2 = m,2, then at this value of Q 2 we have , = 3, 2 = 0.2,


(4.10a)

(4.10b)

(Sv(mt2))2 = 0.

The first condition follows from the fact that no squarks are observed below threshold and the second from previously fitted structure functions below m 2, with m t taken to be 34 GeV (i.e. assuming that the squark mass is 17 GeV, which is its current experimental lower limit). In order to be consistent with eqs. (4.5), (4.6), and (4.9) we parametrize the valence parton distributions as

qvtx.~l= ~q t~/x.,o~,[,og~] "v'°~' 1

Sv(X, Q2)=Cs.(Q2)xA,'Q2)[logx]

"

B%(Q2)

(4.1 la)

,

(4.11b)

where

Bqv(Q2)+ 4CFS (Q2, Qo2 ),

(4.12a)

Bs~(Q2) __ _ Bs~(Q02 ) + 4CFS(Q2, Qoz),

(4.12b)

Bqv(Q2) =

Aqv(Q2 )

Rqv(Q 2) - 2 1 - Rqv(Q2 )

with

qv/o2, (qv/Q2 1)

I/E1 + Bqv(Q2)]

(4.12c)

332

C. Kounnas, D.A. Ross / Hadrons in supersymmetric QCD

Asv(Q 2)

Rsv(Q2)-2 1 - R,v(Q 2)

Rsv(Q2) = ( (Sv(Q2))' (sv(e2))2

with

1/[l + Bsv(Q2)]

(4.12d)

Cq~(Q2) = (qv(Q2)) '

[1 + Aqv(O2)] [,+Bq~(Q2)] r(l + Bq.(Q~))

(4.12e)

'

c,.(e~) = (,.,(e~)), [1 + A,,(Q=)] ''+'-':~''

(4.12f)

r(1 + B,v(e~))

w h e r e (qv(Q2))l,2 a n d (sv(Q2))l,2 are obtained from eqs. (4.5) and (4.6). From this parametrization we can calculate qv(X, Q2) or sv(x, Q 2 ) for any x and Q2 once they are known at Q2 = m t 2.

In figs. 1 to 4 we plot qv(X, Q 2 ) and sv(x, Q 2 ) for three different values of Q2. The quantity qv(X, m 2) is found from the evolution equations of QCD without squarks. In principle this is affected by the possible existence of gluinos and it is for this reason that we plot the distributions for a small gluino mass of 1 GeV (figs. 1 and 2) and for the gluino mass equal to the squark mass (figs. 3 and 4). In practice, it is found that this makes very little difference and that (qv(m2))2 --- 0.21 independent of the gluino mass. We also find that taking A - 0.2 GeV for Q2 between the

A

0.7~-

/ 06 05 04

Valence quark distribution " ..... (",, '... ( , ~ \ .,..

m × : I GeV ........... .,,~a = 50 GeV

\-\~\ ',.

....

~',,',,""'..

"~)

= 2 0 0 GeV

.,~2 =1000 GeV

x '.,

05

,, ...

Q2

\ %~ ......."-,%

01 0

,.~ .-... .... OI

0.2

0.5

04

05

06

Q7

Q8

0.9

X

Fig. 1. Xqv(x, 02) against x for different values of Q 2 in the case where the gluino mass is taken to be l GeV.

C. Kounnas, D.A. Ross / Hadrons in supersymmetric QCD

Valence squark distribution

0 06 0 05

333

m x : I GeV \

.........

,/-O2 = 5 0 GeV

.....

./-O2 =200 GeV

\

0.04 \ 0 03

'"\X -

./-O 2 : I 0 0 0 GeV

o.o

0.01

"" I

0

OI

0.2

0.3

04

05

06

0.7

018

×

Fig. 2. XSv(X, Q2) against x for different values of Q2, in the case where the gluino mass is taken to be 1 GeV.

Valence quark distribution 0.(

"~'\'-.--

m x = 17 GeV

~'~"-, O5 04

"\'k". ""

:

"....

=200GeV

02 -

z

=,ooo

,,, ..,.

02

,, ....

OI 0

~

v,::, \,,

50 GeV

'......... v'Q 2

. , .... OI

0.2

05

04

×

05

0.6

07

0.8

Fig. 3. Xqv(X, Q2) against x for different values of Q2, in the case where the gluino mass is taken to be 17 GeV.

c-quark and b-quark

thresholds

4~r

- 0.012,

for m x = ½mt,

(4.13a)

4~r

- 0.014,

for mx--

(4.13b)

1 GeV.

We also find Bqv ( m 2) = Bsv ( m 2 ) = 2 . 9 4 .

(4.14)

C. Kounnas, D.A. Ross / Hadronsin supersymrnetricQCD

334

l

0 06

Valence squark distribution m x = 17 GeV

005

......... j-Q2 = 50 GeV .....

0.04

~-Q2 = 200 GeV

\

,/-02 = tO00 GeV

0.05 0.02 001 0

OI

0.2

05

0.4

0.5

0,6

07

08

X

Fig. 4. XSv(X,Q2) against x for different values of Q2, in the case where the gluino mass is taken to be 17 GeV.

In the case of standard Q C D without squarks it was found that qv(X, Q2) was Q2 independent at x o = 0.15. Above Q 2 = m 2 this is no longer the case because the integral (qv(Q2))l, is no longer a constant due to the presence of squarks. It is now the sum qv(x, Q 2 ) + Sv(X ' Q2) whose integral is Q2 independent and there exists a value of x 0 for which this sum is constant, although, as can be seen from figs. 1 to 4, this x o has the lower value of x o = 0.05.

5. Singlet parton distributions (qt, st, G, X) Eq. (2.11) is a system of coupled differential equations describing the Q 2 evolution of qt(x, Q2), st(x ' Q2), G(x, Q2) and M x , Q2). Taking the second moment of these equations and using the second moments of Pu shown in table 2 we have (G)2 d Q2 Q_~

(~)2 (qt)2

(s,)2 a( Q 2

8rr

)

- ~c,, - rR

~CA

~CF

2CF

~CF

C~

~CA

-- ~CA -- TR

~r~

~r~

1T R

2T R

~C~ 2C F

(G)2

CF

(qt)2

-4C F

(st)~ (5.1)

335

C. Kounnas, D.A. Ross / Hadrons in supersymmetric QCD TABLE 2

The analytic expression of the anomalous dimensions "ri~and Yi~ n=2

Pdc=( -

Psi= pfl,x = (

2 n-1

2s (.)

2+ 2 n n+l

2 ~) n+2 + 2~-½2~

n+t +

2CA

½"2CA

n+l

2T~

½.2T R

(2 nTl

~2 )2TR nT2

1

1

~.2c~

3~

I

P~x = ( - 2 s ( n ) + . n. . n. + l +~)2CA-22TR

- 34.2CA-- ½.2TR

1 )2TR n+l

~-2T R ½"2TR

P~-q

2

n-I

,]

2

n ~-n+l

4" 2C F

2CF

Pxq=[ ln- n+ll ]2CF n

~6"2CF

1

]

' 2C F

n + l +1 2CF

½" 2C F 2

I • 2C F ½.2C v

½" 2C F P~ = [ - 2 s ( n )

+ i]2C F

- 2 • 2C F

Y i j - fo dx x"-IPij(x ). s(n) is defined as s ( n ) - Y j= jl/j. n

I

½.2~

6~- 2TR

2 2+ 1 ) n~-i--n , + 1 2CA

1 n

z -~ 2~

__

n

•

C. Kounnas, D.A. Ross / Hadrons in supersymmetric QCD

336

The above 4 x 4 matrix has the eigenvalues X0 = 0,

(5.2a)

~kI = -- ( r R -Jr- 3CF),

(5.2b)

(5.2c) with corresponding eigenvectors

(5.3a)

Vo = (G)2 + @)2 + (qt)2 + (st)2, 3C F .

VI m_ ( a ) 2

--1-( X ) 2 --

-~--R ((qt) 2 +

(5.3b)

(st)2) '

V2,3 = (?~2,3- ~ C F ) [ 2 ( G ) 2 - - 8(X)2]

+2CF[](q)2--6(s)2]"

(5.3c)

The fact that the combination V0 has eigenvector zero is a statement of the fact that the energy momentum tensor is a conserved quantity (V0 = 1 for all Q2). The other eigenvalues are all negative so that the other combinations V1, V2 and V3 vanish asymptotically. From this we can determine the fraction of momentum caused by each type of constituent in the limit Q2 ~ oo:

(G(Q2))2

~

4

0.32,

(5.4a)

1 3C F (~k(QE)>2Q2...,oo 5 TR + 3C F

0.08,

(5.4b)

(qt(Q2)) 2 ~

0.36,

(5.4c)

0.24,

(5.4d)

3C F

O2...o o 5 T R + 3 C

Q2~

(st(Q2)) 2 ~

3

TR

5 TR+3C

2

F

F

TR

Q2_,oo 5 T R + 3 C

F

where we have assumed six flavours (T R = 6) and (CA = 3, C F = ~). The above asymptotic results for the second moment agree with those presented by Llewellyn Smith [17]. These asymptotic results should be compared with standard QCD in the absence of gluinos and squarks in which case 0.47 of the momentum is carried by gluons and 0.53 by quarks in the asymptotic limit. Thus we see that for the supersymmetric model a larger fraction of the longitudinal momentum of the hadron

337

C. Kounnas, D.A. Ross / Hadrons in supersyrnmetric QCD

is carried by charged particles (0.60 instead of 0.53). This increase m a y affect the p~ ~ #+~ + X cross section by 10-15%. Putting numbers into eqs. (5.1), (5.2), and (5.3) (for the case of six flavours) the second m o m e n t s of the distributions Q2), ~(x, Q2), q t ( x ' Q2) and Q2) have Q2 evolutions given by

G(x,

st(x,

( G ( Q 2 ) ) 2 = 0.32 + ~ ( G o + )~o - 2) e-10S + ½[0.982(G o - 4),o) - 0.088(2qo - 3So) ] e - ' 6 A s s + ½[0.018(G o - 4)~o) + 0.088(2qo - 3So) ] e 6o4s,

(X(Q2)>2 = 0.08 + -

(5.5a)

(Co + Xo - 3) e - ' ° s

½[0.982(G o - 4)~o) - 0.088(2qo - 3So) ] e - , 6 , 8 s

- ½[0.018(G o - 4)~o) + 0.088(2qo - 3So) ] e 6.o4s,

(5.58)

(qt ( Q 2 ) ) 2 = 0.36 + 3(qo + So - 3) e - , o s + ½ [ - 0 . 1 9 7 ( G o - 4)~o) + 0.018(2qo - 3So) ] e-16.18S + ½[0.197(G o - 4Xo) + 0.982(2qo - 3So) ] e-6.o4s,

(5.5c)

(st ( Q 2 ) ) 2 = 0.24 + ~(qo + So - 3) e - , o s - ½[ - 0 . 1 9 7 ( G o - 47~o) + 0.018(2qo - 3So) ] e - 16-18s - ½[0.197(G o - 4Xo) + 0.982(2qo - 3So) ] e - 6°4s,

(5.5d)

where

Go=(G(Q~))2,

ho = ( X ( Q 2 ) ) 2 ,

qo=(qt(Q~))2, So=(st(Q~))2,

] S=S(Q2, QZ)=½1og 1 + 3a(Q~)l°gQ: 4 - - - 7 - - Q---~o]" 2

F r o m the initial values of G o, X o, qo and s o at the squark threshold Qo2 = m t we can determine the fraction of longitudinal m o m e n t u m carried by each constituent at any Q2. F o r m t = 34 GeV the results of the low-energy p h e n o m e n o l o g y give [12] G o = 0.53,

qo = 0.47,

X o = s o = 0,

G O = 0.45,

qo = 0.47,

~o --- 0.08,

for m x = ½mt = 17 G e V , s o = 0,

(5.6a)

for m x = 1 G e V . (5.68)

C. Kounnas,D.A. Ross / Hadronsin supersymmetricQCD

338

We now look at the large-x distributions for the partons and their Q2 development. From the large-N limits of the moments of Pij shown in table 2 the Q2 evolution for the large moments is given by the uncoupled differential equations (G)N 2 d Q ~

<~>N
N--,~

-

-

8qr

(G)N

0

-4G log N

-- 4 C F

0

-- 4 C F

(qt)N '
which implies for the large-x behaviour of the distributions .

2-[

I ] 4cAs(&, Qo2)

G(x, QZ)x2 G(x, Qo)[logx]

,

~k(x, a2)xZl~k(x, p2)[log~] 4CAS(Q2'Q~), 2-[

1

(5.8b)

]4CFS(Q2'Q~)

qt(x, Q2)x--~lqt(x, Qo)[logx] 2-[

(5.8a)

,

1 ]4cFs(QLQb

st(x, Q2)x___,1st(x, e0)[loG]

(5.8c)

(58d1

To derive the behaviour of the distributions for small x we must examine the leading poles in the Mellin transform, which (from table 2) is at N = 1. In this case we have

Q2

(a)uu-~,

8"I1" IV~--'I (G)N+(}k)N+

((qt)N+(St)N)'

(5.9) whereas the Q2 derivative of the Mellin transforms of all other distributions are regular at N = 1. We seek a parametrization of the parton distribution which is consistent with eqs. (5.5), (5.8) and (5.9) so that they correctly describe the Q2 evolution of the low-x

C. Kounnas, D.A. Ross / Hadrons in supersymrnetric QCD

339

limit, the large-x limit and the fraction of momentum as carried by each constituent. We begin by decomposing the quark and squark distributions into "valence" and "sea" parts:

qt(x ' Q2) = qv(X ' Q2) + qs(X ' Q2),

(5.10a)

st(x ' Q2)= Sv(X' Q2) + Ss(X' Q2).

(5.108)

A parametrization of the patton distribution consistent with eqs. (5.5), (5.8) and (5.9) is given by

X(x, Q2) c~(o ~) r, 11"~Q2~ x

[t°gx]

'

[

(5.11a)

qs(X,O 2) Cqs(Q2) log

,

(5.11b)

ss(x,Q z)

,

(5.11c)

X

C~'(Q2) log X

[, 1 ] 8qv(&) qv(x, Q2)=Cqv(Q2)x~qv(Q2)[lOgx] , "

1 1 Bs,,(Q2)

Sv(X, Q2)=csv(Q2)xA~'&)[logx] G(x, Q2)

CG(Q2) RG(x,

(5.11d)

,

(5.lie)

Q2)+ Cx(O2--------~) Rx(x, QZ)-)t(x,Q 2)

X

X

+ ~CF A Cq~(Q2)xRq(x'Q2)~

Css(O2) , x Rs(X'Q2)

-qs(x, Q2)-ss(x, Q2)},

(5.110

where

R,(x, Q2)=r(I+B,(Q2))

log(l/x)

a,(O 2)

Bi(Q2)/2

I.,,Q:)(2~ai(Q2)logl/x). (5.12)