Nuclear Physics B104 (1976) 445-476
© North-Holland Pubhshmg Company
HEAVY QUARK CONTRIBUTIONS TO DEEP INELASTIC SCATTERING *
Edward WITTEN
Joseph Henry Laboratortes, Princeton Umverstty, Princeton, NJ 08540
Recewed 17 November 1975
We show that m an asymptotically free gauge theory, the contnbutaon of heavy
quarks to deep inelastic structure functions can be computed m terms of the contributions of hght quarks In the standard model with four quark multaplets and an SU(3)
gauge group, we estimate that for l0 GeV2 < Q2 < 40 GeV2 the charmed quark contnbutlon to structure functions wtll be several percent for x < 0 2 and tenths of a
percent for x > 0 2 Our main theoretical tool ~s an expansion, analogous to the usual
operator product expansion, of an operator constructed from heavy quark fields as a
sum of hght quark and gluon operators with c-number coefficients This expansion Is
vahd order by order m perturbation theory
1. Introduction
Many o f the most excmng results m high.energy physics m recent years have
come from measurements o f deep Inelastic lepton-nucleon scattering The success
o f the naive parton model [1 ] strongly suggests that the nucleon can be regarded
as a bound state o f pomt-hke, charged constituents Within the context o f asymptotically free gauge theories o f the strong mteractlons, it is possible to justify many
o f the parton model results and to make many new predictions, mostly concerning
scaling vlolatlons at high Q2 [2]. In this paper, we will assume that the strong rateractions are in fact described b y a non-Abehan gauge theory o f quarks and gluons,
and we wdl further assume that, m addition to the three famlhar multlplets o f p,
n, and ~, quarks, there is at least one ad&tlonal heavy quark multlplet, perhaps the
charmed (p') quark o f a popular model o f the weak and electromagnetic interactions [3]
A remarkable feature o f deep inelastic scattering Is the suppression o f the quarkantlquark "sea" The most sensmve measure o f this, from existing data, comes
from the success o f the naive quark model relation a v = 3o F for high-energy n e u t n n o
and anti-neutrino scattering If q(x) and ~(x) represent the quark and antlquark
* Work supported m part by the National Science Foundation under Grant Number MPS7522514
445
446
E ICltten / Deep melastw scattenng
momentum &stnbutlons of the nucleon *, then one finds from present data
1
f dx~(x)
0
1
=005+002
f dxq(x)
o
Moreover, nine tenths of the total contribution to f01dx ~(x) comes from the region o f x less than 0 2 [4] If we assume that there are at most no more X or ~'
t
--t
antlquarks than p or n antlquarks, and that ~ X, p ~ p , we conclude that
foldxCh + X') ~< 0 04, foldx(p ' + if') ~ 0 04 On this basis it appears that 90% of
the quark distribution of the nucleon consists of the naive p and n quarks, with
the &screpancy confined almost entirely to the region x <~ 0 2
The only clear-cut reason known, w~thm the context of gauge theories, for the
suppression of the "sea" is the 1/N expansion As shown by 't Hooft [5], m a
theory with gauge group SU(N), a &agram with a quark loop is suppressed by a
factor o f N relatwe to a slmdar diagram with the quark replaced by a gluon The
antlquark and strange and charmed quark content of the nucleon is presumably
associated solely with &agrams with quark loops If the gauge group of the strong
Interactions Is SU(3), we would thus expect the "sea" to be suppressed by a factor
of three Although this must represent a partial explanation, the suppression of
the quark-antlquark sea, especially at large x, is much greater than a factor of three
In ttus paper we will consider the only feature of this problem which appears
to be tractable at present from a theoretical point of view This is the problem
of the &stnbutlon m a light quark state, such as the nucleon, of heavy (charmed)
quarks [6] We will see that m an asymptotically free theory, the problem of the
heavy quark content of hght quark states is soluble In the hmlt as the heavy quark
mass becomes mfimte, ~t is possible to calculate the &stnbutlon of heavy quarks
in any hght quark state m terms of the distribution of hght quarks m that state
The mare results are as follows m measurements o f F 2 for 8 GeV2 ~< Q2 ~<
40 GeV 2 we expect the charmed quark to be present at the level of several percent
for x ~< 0 2 and at the level of tenths of a percent for x > 0 2 In measurements of
F 3 we expect the charmed quark to be nearly absent (suppressed by a factor of ~,
the effectwe couphng constant, relatwe to the charmed quark contribution to F 2 )
The longitudinal structure function F L is, of course, very small, we expect the
charmed quark contribution to F L to be 10% to 20% of the total Finally, the
charmed quark and charmed antlquark &stnbutlons should be equal to within an
error of order ~ The precise numbers are dependent on Q2, on the current con-
* In other words, q(x) is x times the probability to find a quark carryinga fraction x of the
nucleon momentum
E ICttten /Deep melasttc scattenng
447
sldered, and on assumptmns concernmg the masses and scale parameter, in a fashmn
that wall be described
If the heavy quark were sufficmntly heavy, it would be posslble, m principle, to
subject non-Abehan gauge theories to a rigorous experimental test by measuring
the charmed quark contnbutmn to structure functions If the mass of the heavy
quark is actually only 1 5 - 2 GeV, as is suggested by the mass of the new resonances, we expect most of our predictions to have a large uncertainty, perhaps a factor
of two
In sects 2 and 3 we discuss the concepts that are required to evaluate the
charmed quark contnbutmn to deep inelastic scattering m the regmn Q2 >>4M2,
were M is the heavy quark mass We include a brief &scusslon of recent work by
Appelqulst and Carrazone [7] In sect 4 we consider the ideas needed to extend
our results rote the region Q2 ~ 4M 2 ' most of the exastmg and foreseeable experimental data are in this regmn The remainder of the paper is concerned w~th quantitative estimates
2 Matrix elements of heavy particle operators
In the short &stance analysis of deep lnelastm scattenng, one encounters operators of twist two and arbitrary spin constructed from quark and gluon fields Information about their matrix elements can be directly translated rote predlcUons concernmg the deep inelastic structure functions Our first task is, therefore, to determme how to calculate the matrix element of a heavy quark operator
Actually, we will prove a statement that would be vahd m any renormahzable
field theory with more than one scale of masses For slmphclty, consider a scalar
field theory with one heavy partmle ¢ha and one hght particle ¢~1 The Lagrangmn Is
L = ½(d.~bh)2 + ½(d.~l) 2 - LM2q~22 h -- -~m2dp? -- ¼g(qb2 + ~?)2
Now we consider the heavy particle operator of spin n and lowest twist
on,.
h ~ "n(x) = d~h~[. 1
gUnCh(X) -- trace terms
This operator has the smallest canomcal dimension of all of the heavy partmle
operators of spin n, that is, operators whmh are traceless and symmetric in n Lorentz
radices We are interested m matrix elements of this operator between states constructed from hght particle fields at low momenta In a crude sense, this measures
the probabdlty to find a heavy particle present m the hght particle state
We wlsh to prove that this matrix element, m the hmlt M ~ 0% is simply equal
to a c-number functmn A(g,M.m,#) times the corresponding matrix element of the
hght particle operator with the same quantum numbers
OI(X) = ~51~.1
~lan~)l(X)
The functmn A does not depend on the physical states considered, but only on the
448
E Wltten / Deep melastw scattermg
couphng constant, the masses, and (depending on the renormahzatlon procedure)
the subtraction point/a (We will see that if the mass-mdependent subtraction
scheme of Welnberg and of 't Hooft and Veltman [8] is followed, then A is rodependent of hght particle masses.) The corrections to our formula, m any fimte
order o f perturbation theory, are of order 1/3/2 (times a possible polynomial m
logM)
The result is reminiscent of the operator product expansion, and the argument *
is a matter of power counting To begin an mductwe argument, consider the lowest
order contnbutmn to the two and four hght particle matrix element of Oh, shown
m figs 1 and 2 To this order, these are the only nonzero hght particle matnc elements of O h
Fig 2 has no ultra,aolet divergence and, because of the propagators 1/(p2-M2),
it vamshes hke 1/M 2 m the large M hmlt Fig 1 is ultravmlet dwergent, so it is not
possible to remove factors of I[M 2 from the propagators After being rendered
fimte by a subtractmn, fig 1 does not vamsh as M ~ oo
However, suppose that we dffferentmte dmgram (1) n + 1 times with respect to
external momenta (recall that n is the spm of the operator) It is now convergent
by power countmg, and hence, because of the propagator factors, it vamshes hke
1//i/2 Therefore, lgnonng the term that vamshes hke 113/2, fig 1 Is equal to a polynomml m the momenta o f degree at most n, times a momentum-independent functmn A(g,M,m,#) The polynomml is a traceless, symmetric Lorentz tensor with n
m&ces, so (except for a constant factor that can be absorbed m A) it must simply
be equal to (29 + p')~l
(p + p')~n _ trace terms, which is simply the free field
theory matrix element of the hght particle operator O 1 between the same states **
Thus, to this order, we have estabhshed that for some smtably chosen c-number
functmn A, the formula O h = A O 1 is vahd for all low momentum, hght particle
matrix elements of O h
Now suppose the substltutmn formula has been justified and the functmn A
constructed to order gk In order gk+l we encounter, first of all, various matrix
elements of O h w~th more than two external particles These matrix elements are
convergent by power counting once all lower order counterterms have been subtracted, and they have skeleton expansmns
Consider now some term m the skeleton expansmn of such a convergent matrix
element Subtleties involving ultravmlet &vergences are buried reside the defimtlon
of the dressed propagators and vertices that appear m the skeleton Hence xf the
* For a similar argument m connection with a somewhat similar problem, see ref [9]
** More precisely, the possible appearance of terms proportional to (p - p')# is irrelevant,
because we are interested m apphcaUons to the case of forward scattenng, p = p' The
terms proporUonal to (p - p')# correspond to the appearance m the expanston of additional operators, such as
~(~, ~2
~n~),
whose matrix elements vamsh at zero momentum transfer
449
E lCttten / Deep melastw scattenng
,
I \
Fig 1 A contribution to the two hght particle matrix element of a heavy particle operator
Dashed lines are hght particles, sohd hnes are heavy parUcles, the x is an msertaon of a heavy
partacle operator
Fig 2 A contribution to the four light particle matrix element o f a heavy particle operator
heavy particle Is explicitly visible in the skeleton, the entire diagram is o f order
1/M 2 We may therefore assume it does not appear
Somewhere in the skeleton there appears, then, a light particle matrix element
o f O h constructed to order gk or lower Wherever it appears, we may, b y the induction hypothesis, replace O h b y A O 1 But this simply yields A times the correspondmg term m the skeleton expansion o f O 1 This establishes that the rule
O h = A O 1 Is valid to order gk+l for convergent matrix elements o f O h
It remams to consider the two particle matrix element o f O h This object is
logarithmically divergent b y power counting, and so has no skeleton expansion
However, if we differentiate with respect to external momenta n + 1 times, so as
to obtain an object which is convergent, we obtain an object with a skeleton expansion to which the preceding argument can be applied It follows that the rule
O h = A O 1 is valid for the two-pomt function in order g k + l , except for a possible
polynomial o f order n that was lost b y differentiating As before, this polynomial,
since it is traceless and symmetric and has n Indices, must be proportional to the
free field theory matrix element o f O 1 Consequently, the new polynomial term
can be absorbed in our expansion b y adding a new term o f order gk+l to the defiration o f A This completes the inductive step
We can use the same argument to see that, in the mass-independent subtraction
scheme, the function A is Independent o f light particle masses In fact, In the massIndependent subtraction scheme, all matrix elements o f O h, when differentiated
with respect to m, become convergent, and have skeleton expansions, and the
previous argument can be repeated to show that A is independent o f m
We must add a few words concerning the extension o f the above result to more
general situations In general there might be not just one light particle operator O 1
o f spin n and twist two, but several In flus case, the same argument can be used to
prove the existence o f an expansion O h = 2; A~6~I, where the sum rules over all light
particle operators with appropriate quantum numbers
For applications to a gauge theory, we will usually take O h to be the gauge mvariant operator
S~h?ulDu: z
Bt~n 1,0h -- trace t e r m s ,
where S represents symmetnzatlon and D is the covarlant derwatlve There are two
E Wztten/ Deep melastw scattermg
450
relevant gauge invarlant light particle operators fl')'uiDu2 Dt~n~l and the gluon
operator SFaziDt~ 2 Dt~n- ll~n , trace terms However, as in the case of the operator product expansion, non-gauge lnvarlant operators of twist two, such as the
ghost operator dp*d~i dunce,will also appear in the expansion We will proceed
on the assumption, which has been partially justified in the case of the operator
product expansion [10], that m constructing the physical consequences of our
expansion, we may simply ignore the occurrence of gauge variant operators
For applications to operators of twist two in a gauge theory, we expect the corrections to our formula to be of order 1/M 2 Correctmns of order 1/311are forbidden
under the symmetry ~h ~ "}'5~h, M ~ - M For operators such as ~hlkh which are
odd under the symmetry, it is easy to see that corrections of order 1/3f appear already at the one-loop level
We actually wish to apply our results to amplitudes involving physical hadrons,
which we regard as composite But any amplitude lnvolvang a proton, for example,
can be computed by calculating an amphtude involving two external p quarks and
an external n quark, and looking for an appropriate pole Moreover, one may choose
the p and n quark momenta to be of order one third the proton mass For such an
amphtude our result about the matrix elements of heavy particle operators has been
established We thus expect it to be valid when the target is any of the light hadrons
3 The effective field theory at low momenta
Appelqulst and Carrazone [7] have shown that, m a field theory with hght partlcles and a heavy particle of mass M, the heavy particle decouples from light particle physics, except for renormahzatlon effects and corrections of order 1/M2
Indeed, one would suspect intuitively that this must be so Otherwise it would
be possible with lab experiments at low energies to determine the existence or nonexistence of quarks of arbitrarily high mass
There is a simple corollary of the result of Appelqulst and Carrazone that we
will need Before stating it, we must give a precise statement of their result
Consider again the model Lagranglan of sect 2
L =
1 (d
2+
2 -
~ M 2 ~ _ 1 m2~b2 -
¼g(
+ F) 2
Alongside it we may also consider the modified Lagranglan L* with the heavy
particle and its couplings omitted
L* = ½(du~l) 2 - ½m2~l2 - ¼g ~4
Now, let Fn(g, la,m,M,pz) be the n light particle proper vertex derived from L,
using the mass-Independent subtraction scheme with scale parameter/a, and let
P* be the n-particle proper vertex derived from L* Then F* must equal F n, to
within an error of order 1/M2, provided that the couphng constant, the mass, and
the scale of the Green functic, ns are suitably redefined That is, there exist certain
E Wltten/ Deep melasttcscattertng
451
functions g*(g,M, la), Z(g,M,#), and m*(g,M,m,la) with
I'n(g,M,m, la,p,) = Zn(g,M, Is) I'*(g*(g,M, la), m*(g,M,m, la)~a,p,)
(1)
(Our supposlhon that g* and Z are independent of m is justified by the fact that
they can be evaluated by studying the Green functions m the asymptotic region,
where they are independent of m ) This xs the result of Appelqmst and Carrazone
We will find useful the equation for g* that can be derived by applying the
homogeneous renormahzatlon group equatlon to (1) Both I"n and P* sahsfy homogeneous renormahzatlon group equations [8], although the coefficient functions
differ
/a~-~+
+5
+m
-n7
Pn=0,
(/a d-~ + 3 . ddg* +6*m~-~-~ - n T * ) I'* = 0
(2)
(3)
If one apphes (2) to (1) and requires that the resulting equation for F* should
coincide with eq (3), one obtains the following condition for g*
3,(g,)=(/ad+
d +
dl~ 3-~
8M-~-)g*(g,M,l~)
(4)
It is flus equation that we will need
4. The generalized Wilson expansion
The preceding discussion will enable us to calculate the charmed quark contribution to structure functions in the asymptotic regmn m which the operator product
expansion is vahd and the mass corrections are negligible The parameter that must
be small is 4M2/Q 2 (at least for charm conserving currents which must produce
heavy quarks in pairs) At Q2 of 20 GeV2 this condition is just beginning to be
satisfied Smce much of the relevant data, especially at small x where one may hope
to see charmed quarks, is hkely to be at smaller values of Q2, it is worth whde to
develop a simple alternative approach which is vahd when 4M2/Q 2 Is of order one
In studying deep inelastic scattering one is mterested m the amphtude for the
forward scattering of a "current" by a hadron target Welnberg's theorem [11 ]
ensures that for Q2 much larger than the masses and the target momentum, this
amphtude is dominated by diagrams m which a minimum number of quark and
gluon hnes connect the "large Q2,, part of the diagram, containing the two current
operators, with the remainder of the diagram
Now suppose that Q2 IS much larger than the masses and momenta of the external particles, but is of the same order of magmtude as the mass of the heavy
quark, that is, the mass of one of the fundamental fields
E Wztten /Deep melastw scattenng
452
In this case, by power counting, the amplitude is dominated by diagrams in
wtuch a minimum number of light particle hnes connect the two currents with
the remainder of the diagram Other diagrams have a greater number of propagators
far off mass shell and uncompensated by loop integrations
An lteratlve argument [12] similar to the one given in sect 2 now suffices to
estabhsh that the current scattering amplitude Is equal to a c-number function,
independent o f the target state, times the matrix element in this state o f some
local operator Tlus is the statement of the operator product expansion. The presence in the theory of a heavy particle with Q2 comparable to M 2 modifies the
operator product expansion, and its derivation, only in that we must now expect
that the coefficient functions in the operator product expansion will depend on
M 2 as well as on Q2 and on a mass or scale parameter characteristic of the light
quarks
Since a detailed argument would be somewhat repetitious of previous arguments in this paper, we will simply illustrate the point with an example
Consider a simplified model of the weak interactions with a light ;~ quark and
a heavy p' quark First consider the forward scattering of a W+ boson by a ;k quark
in tree approximation (fig 3) The propagator contains a denominator
1/(q 2 + 2q k + k 2 - M 2) (The propagator numerator, which makes no essential
difference in the discussion, will be ignored for simplicity ) In this order, the usual
operator product expansion consists of lgnonng k 2 and M 2 relative to Q2 and expanding in powers o f q k/q 2
co
1
q2+2q
k
q2
=
(
-
(5)
In tree approximation the matrix element of the operator ~,~,ulDu2
quark state is kvt
kvn and the expansion (5) can be rewritten
Dun~, In a ;k
oo
1
q2+2q.k
_ 1 ~
(_.~2)nq.ul
q2 n=O
q~n(o~ 1 ~n)
(6)
t
X
Fig 3 The tree approximation to the seattermg of a W boson by a ~. quark The wavy hne is
a W boson, the dashed hne, a h quark, the sohd hne, a heavy quark
Fig 4 A contribution to the scattering of a W boson by a gluon (the loopy hne represents a
gluon)
Fig 5 A contribution to the two gluon m,ttrtx element of
--~-~¢vlD/z2 BUnk
E Wttten/Deep melastwscattenng
453
where (Oul .Un>Is the matrix element of k~'ulD~ . D~nk The sum over n corresponds to a sum over t-channel angular momenta
Now clearly, if we choose to consider 3/2 as comparable in magnitude to Q2 but
to ignore k 2 relative to M "2 and Q2, we can say
1
q2+2q.k+k2-M 2
q2 - 21,/2 n~--0
1
q2+2q
q2
k-M 2
n
q
(0~,I Un>
(7)
This estabhshes that m tree approxamatlon the operator product expansion remains
vahd when there is an internal mass of order Q2, the effect of the large mass of the
charmed quark is to modify the coefficient function
q2
in (6) to
1
m (7)
Now consider, m one-loop order, the forward scattenng of a W+ boson by a gluon
One diagram Is shown m fig 4
The upper half of this diagram is the W + - k scattenng diagram that we have already considered in fig 3. We know that (7) is a good approxtmatlon to this diagram if the k quark momentum is sufficiently small
If the upper half of fig 4 can be replaced by (7), we would obtain simply the
sym over n of Cn times that contribution to the matrix element in the gluon state
of ~TutDu2 . Dunk which is obtained by shnnkmg the upper half of fig 4 to a
point, see fig 5
As m our previous discussion, this replacement is not possible because the ultraviolet convergence is not good enough. But for any gwen value of the t-channel
angular momentum, if one differentiates often enough with respect to the external
gluon momentum, one improves the ultravxolet convergence enough as to make
tlus replacement possible The discrepancy is therefore a polynomial in the momenta and thus Is proporl~nal to the free field theory matrix element in a gluon
state of the operator Ful aDm • Dun-1 l~un"By Incorporating a suitable term proportlonal to this operator, with a coefficient function of order g2, one ensures
that the operator product expansion is valid to this order
A more graphic description Is the following Fig 4 has a loganthmlc dependence
on k 2 (or on the light quark mass l f k 2 is set equal to zero) Fig 5 has a similar
logarithmic dependence on k 2 or m 2 We must claim that for given t-channel
angular momentum, the coefficient of the logarithm in fig 4 Is equal to Cn times
454
E Wltten / Deep melastw scattermg
the coefficient of the logarithm m fig 5 We have venfied exphcltly that this is so,
the coefficient of the logarithm m fig 5 is *
g2T(R)(n2 + n + 2)
41r2n(n + 1)(n + 2 ) '
and m fig 4 the coefficient of the logarithm in the nth partial wave is
g2T(R)(n2 + n + 2)
1
[
47r2n(n +l)(n + 2) (q2 _ M 2) ~ q2
2 in
MJ
Thus, after subtracting Cn times the matrix element of ~TulDu2 DunX from the
nth particle-wave contribution to fig 4, one obtains a local object, a polynomml
in k The coefficients of these polynommls, which are comphcated functions of Q2
and M 2, have some phenomenologlcal interest and are discussed below
To prove the vah&ty of the operator product expansion to all orders is simply
a matter of repeating and systematizing these considerations
5 Renonnahzation group analysis
Let us now consider a renormahzat~on group analys~s of the foregoing In tlus
section we consider only the large Q2 region, Q2 ~ 4M 2
The operator product expansion enables us to expand the product of weak or
electromagnetic currents as a sum of c-number functions times operators of twist
two If the number of hght quark multlplets is equal to r, then the physics of the
hght quarks has an approximate SU(R) symmetry, and we wall find it useful to
first classify the operators m the short distance expansion according to SU(R)
Correspo__ndmgto spin_n, thereappear in the expansion threeSU(R~smglet
operators ~hTulDu2 Du n ~h, t.kzTulDu2 Du n ~kl, and FaulDu2 Dun - 1F~n
(subscript t, I, k will denote sums over hght quarks only, subscript a, b, c wdl denote sums over all quarks) In view of the analysis in sect 2 of this paper, it is clear
that low momentum matrix elements of the first of these operators can be computed m terms of matrix elements of the last two, and this wdl enable us to make
some new statements concerning deep inelastic scattering The SU(R) non-stagier
operators, however, are constructed from hght quark fields, so we have nothing
to say about their matrix elements
The current product which, m a parton model sense, measures the abundance
of heavy quarks m the target is the product of heavy quark currents,
¢JhTU~bh(X) ~0h~,V~kb(0) The current product which, m the same sense, measures
the total abundance of quarks m the target is the product of baryon number currents, ~a7 u~ka(x) ~bTv~kb(O) We wall attempt to calculate the short &stance,
* Of course, this number follows from the calculation of anomalous &menslons m ref [2]
The numbers gwen m the text actually include the contnbunon from the crossed graph
E ~tten / Deep inelasticscattenng
455
heavy mass limit of the ratio of the matrix elements of these current products,
and thereby, to determine the fraction of all quarks in the target which are heavy
The quantities which are actually physically relevant are
fd4xetq XT(~h'ytZ~h(X ) fh"/V~bh(0)) ,
which we will denote as JJh(q),
fd4xetq XT(~a'yU~Oa(X) ~bTV~b(O)),
which we will denote simply as
SJ(q)
For the short distance analysis, we must expand in operators of definite anomalous dimension, which will be those with definite transformation properties under
SU(R + 1) Corresponding to spin n we consider the SU(R + 1) non-smglet
-
n
-
ffqJNS = ~bhl'ut u2
D~ ~ h -"~-~"~~aT~iB ~
B~n ~a
and the slnglet operators
~ll/n = ~//--a,Y/zlO/~2
D~n~/a
FF n = o#~iD~2 D~n_ib'~n
Henceforth the superscript n IS suppressed and in the discussion of the operator
product expansion, a sum over the spin of the operators, which ranges from two to
Infinity, is understood but not explicitly stated
According to the operator product expansion, there exist certain coefficient
functions A, B, and C, with
JJh(q) =A(q2da,g) ~t~Ns + B(q2~,g)( ~F),
(8)
JJ(q) = C(q2,1a,g)( ~F~F) ,
(9)
Here B = (B1,B2) and C = (C1,C2) are two component row vectors /a is, again, the
scale parameter of the mass.independent subtraction scheme
In the effective theory defined by omitting the heavy quark and its couplings,
there are two SU(R) smglet oj~erators of spin n and twist two, namely
B~n ,l~l and FaulDta2 . Dun_llT~n We will denote these as ~ * and
FF*
From the discussion In sects 2 and 3, we know that for purposes of low momentum matrix elements, the operators of the full theory with parameters g, M, and g
can be replaced by a sum of c-number functions t~mes the operators of the effective
theory with parameters g*(g,M/#) and g Thus,
~l'~glOu2
~NS(g,M,/I) =
U(g,M,la) \ FF* ] (g*ta) ,
(10)
E h~tten/Deepmelastwscattering
456
(~F~F)(g,M,I~)=V(g,M,I~)(~F~)(g*~)
(11)
Here U = (U1,0"2) is a row vector and V is a two.by-two matrix
Combining the above, we determine that
It is now straightforward, although somewhat te&ous, to carry out a renormahzatlon group analym and to obtain a useable formula relatmg IJh(q) to JJ(q)
The operators m the short distance expansion sahsfy the equations
'NS ¢'NS =o
=o
Here 7(g) and 3'*(g*) are two by two anomalous dimension matnces
In the standard fastuon, one derives renormahzahon group equations for the
coefficient functions A, B, and C by applying the renormahzahon group operator
D = (pd/d/z +/3d/dg + 8Md/dM) to (8) and (9) and using the fact that the currents
are unrenormahzed We obtain
+~
+ 8M~-
d
TNS A = 0 ,
C =0
(12)
To obtain renormahzatlon group equations for U and V we apply the renormahzanon group operator to (10) and (1 t) Making use of the fact that U and V depend
only on g and the ratio M/N, we obtain
( - ( 1 - 8)3/--~ + j 3 ~ g ) U + 3'NSU -
U't*(g*)=O,
( - ( 1 - 8)M~-~ + ~-~-) 1/'+"/V- VT*(g*) = 0
(13)
E Wztten/Deep melasnc scattenng
457
The solution to (12) is standard
A(q2~g)=exp(f
dt ~'NS~
t) ) A Cq 2 ,1a,g Cq 2)) ,
~--ff~
~-(q2)
B(q2jt.g)=B(q2dt.,)Texp(f
dt~),
~-(q-)
C(q2,1a,g) = C(q20,la,g ) "~
. e x p [[j .g_
~t~/(t)\
u
?(5)
f(q)
(T and ~ wdl represent hme ordenng and anti-time ordenng, respectively ) The effective couphng constant has been defined m the usual fashion q2(d/dq2)~q2.g) =
I~(g--).with the boundary condihons ~q20.g) =g
The solution of (13) Is
/ g
U0~M,,)=exp(f
dtTNS(t)\u( M, ~(M 2~
~-~3--) C~ o,~, -r
exp'f(g'M)
7*(t)~
(,,~_(Mb~o,d, ~-~-/y/,
F(M ~ )
V(fl,m,g) = r exp(- fg
7(0
Mo,g())
- M2 ~
dt ?~)V(~
~(M')
1 g*tg,M)
X exp~f
2~,o ~
dt
~*(t)~
(14)
~-~!
Here g(M"2) satisfies Mdg/dM =/~(g-)/(1 - 8(g-)), ths deviates shghtly from the usual
equation defining the effectwe couphng constant, but we wdl find the difference
ummportant Verifying that (14) samfies (13) is straightforward, one must make
use of eq (4) from sect 3
Now we can put all the pieces together, and we learn
g(M2)
2 - 2
7NS(t)
JJh(q)=A(qo,la,g(q )) exp(f
dt ~--ff~)
- ~(q2)
...
.
x U(U,Mo,g(M2))rexp( f
g*(g,M)
7*(t)~/~O*~
dt-uo-y]IFF.]
g*(g-(M~),Mo)
+B(q2da.,(q2)) ~ e x p ( / ( M ' ) - 7(O \
f(q2) d t ~ )
X ~.0.,..~..
~ exp(/"~'~
dt~)
g*fg(M¢).Mo)
(~;),
E Wttten/Deep melastwscattering
458
JJ(q)=C(q2,g,~(q2)) Texp( /(M2) dt T(t) ~
g(q2)
B(t) ]
g*(g,M)
X V(la,Mo~(M2))Texp( ["
\d
~
,
- ,
dt~(
~g/ ~
g it) ]\FF* ]
(15)
g*(g-(M ),Mo)
For M 0 of order ta, g*(g-(M2),M0) -~ ~M2), whereas g*(g,M), the physical coupling constant of the effective hght quark theory, is presumably of order one
Since all effective couphngs are small, the only uncalculable factors m (15) are
the common factors wluch appear on the extreme right Therefore the ratio of
JJh(q) to dJ(q) Is completely calculable. Since this IS rather comphcated, we discuss
for the moment only special hmmng cases
The free field theory values of the various coefficient funcUons are
A = a 0 (a constant whose value IS Irrelevant),
B = (ao/(r + 1),0),
c = (ao,O),
U = ( - 1 / ( r + 1),0),
° 0)l
If we now Imagine that M/la and q/M are sufficmntly large that our formulas are
vahd, but that g~q2) is not too different from AM), then we can ignore the factors
/ g'(M2)
3,NS(t) J~,
oxplf ' dtg(q)
~ e x p ( f ( M 2 ) dt 7(t) ]
gCq2)
~(t) 1'
m (15) Inserting the free field theory values of the coefficient functions in (15),
one finds JJh(q) = 0 Thus, this IS the limit in which the heavy quark would decouple from deep inelastic scattering
On the other hand, in the truly asymptotic region, q ~ oo with M/p large but
fixed, the first term m the expression above for JJh(q) can be dropped, since "rNS
is larger than the smallest elgenvalue of % and ultimately the suppression associated
with this factor dominates everything else Therefore, In the true asymptotic region,
JJh(q) andJJ(q) differ only by the free field theory rano orB to C, which is 1/(r+ 1)
This is m agreement with the result, well-known from previous work [2], that In
the true asymptotic regmn the cross section is pure SU(R + 1) slnglet, since the
smglet operator has the lowest anomalous dimension
We will discuss later a more detailed quantitatwe evaluation of (30)
E lCttten/Deep melasttcscattermg
459
6 Analysis of the "small Q2,, regmn
Before proceeding with a renormahzatlon group analysis of the charmed quark
contribution to deep Inelastic scattering in the region Q2 ~ 4M 2, we will find it
useful to first consider what we mean by the mass of the charmed quark
If In some fashion an effectwe mass M* of the heavy quark is determmed by
experimental criteria, this effective mass will be Independent of the arbitrary choice
of a subtrachon point and so will satisfy a homogeneous renormahzatlon group
equation
The solution is
g rltS(t)~
M*(g, la,M/la) = (M/g) M*(~(M)da,1) e x p [_
, - _/ ~./~(t) J
~-(M)
(16)
In a non-confining theory, with physical charmed quarks assumed to exist, we
would slmply define M* to equal the physical charmed quark mass The zero couphng limit ofM*(g, la.M/la) would then simply equal M, and from (16) we would
learn that for large M,
g
M*=Mexp(-f
8(t)~
(171
k(M)
We claim that it Is M* rather than M which should be identified as the phenomenologlcal charmed quark mass and should be estimated as ½ the mass of the J As
evidence for this we observe that, as we have just shown, our identification would
be correct m a non-confining theory, and that it is M* rather than M which is renormalization group Invanant and therefore has some posslbihty of being observable
214" is the quantity whmh appears naturally m our calculation and m other renormahzatmn group calculations of heavy quark properties
We now return to the renormahzahon group analysis We have at our dlsposal
an expansion of the current products m terms of c-number functions multiplying
the local operators of the effective theory with the charmed quark omitted
JJh(q) = G(q2,M,mg) ~ FF* ] '
JJ(q)= H(q2,M, la,g) ( ~F~,)
In contrast to the previous, large Q 2 case, G and H do not factor Into the product
of a function independent o f M 2 times a function Independent of Q2. To obtain
information about G and H, one uses the renormahzatlon group to scale down both
Q2 and M 2 until both are of order/a 2
E IVttten / Deep melasttc scattenng
460
Analogous to (12), G and H now satisfy
(7
,
,
The solution is
(q2,g,~l,§(q2,g)) T e x p
dt~
g*ff(q~))
Here
/ q02xl/2
/
g
IS the "effective mass' m the sense of Wemberg and 't Hooft [8]
In the regmn m which the calculatmn described here is of interest, Q2 is comparable m magmtude to ?/2, and we may neglect m the expressmn for M the difference between ~(q2) and g(M 2) Hence M = M*(q2/q2) 112, where 34* is the
quantity (17) wtuch we claim Is actually observed as the phenomenologlcal quark
"mass". Our final result is thus
G (q 2 41"M'g)= (GH)(q241"M*(q2/q2)l[2"(q2))
(H)
/ g*¢g)
x
exp(f
2))dt -q61
(18)
It ~s usually most convenient, of course, to take q2 = ~t2
I f ¢ 2 ( q 2) and ¢2(q2) In (Q2/M2) are both small, then the quantttles G and H
appeanng on the right side of (18) can be calculated reliably m lowest order perturbaUon theory This Is the condition under which (18) is useful By comparison,
expression (15) denved m the previous section is useful l f ¢ 2 ( q 2) and 4M2/Q 2 are
both small
7 Phenomenological applications
We are finally prepared to make quant~tatlve estimates We consider m turn the
standard structure functions F 2, F L = F 2 - 2xF1, and F 3 , as defined by
1 fay e'. Y<par(J.(y) J,(0))lp>
_ PI~P~;'2(q2,x)
mq p
gt~Fl(q2,x) ÷ let~uorpOqrF3(q2'x)
m
2mq p
'
E l~tten / Deepmelastwseattenng
461
where p is the m o m e n t u m o f the target state Ip) and x = -q2/2q p
The renormahzatlon group analysis yields, o f course, most dlrectl3~ the moments
o f the structure funcUons [ 13] The operators o f spin n determine the moment
f01 dx xn-2F, where F l s F 2, F L , or xF 3 We will find that a rather clear picture
o f the charmed quark & s t n b u t l o n can be obtained b y a consideration o f the specml
cases n = 2 and n = 4 Thus, we wall calculate the quantities R 2 = (f01dx FH)/foldx
and R 4 = (f01dx X2FH)/(foldx x2F), where F represents some structure function
of the baryon number current and F H Is the contribution o f the heavy quark to
this structure function
71F 2
In the parton model sense, F 2 is simply equal to the quark m o m e n t u m distribution [1 ]
Actually, m the region Q2 ~ 4M 2 it Is necessary to specify which o f several possible current products we are using to determine the heavy ~quark content o f the
target the product o f heavy quark vector currents, the product o f heavy quark
axial vector currents, or a product o f charm-changing currents, such as ~3,up'p'TvX
We will give numencal results for the charmed quark contributions m each case
In the asymptotic region, Q2 >> 4M 2, the SU(R + 1) X SU(R + 1) ctural symmetry o f the operator product expansion ensures that ~t does not matter which
current one uses to measure the heavy quark content o f the state
We consider the "small Q2,, region first
The expansion o f the product o f two baryon number currents in terms o f local
operators, contains, m zeroth order m the effective couphng constant, a term proportlonal to the fermlon operator ~tT~lDu2
D~n ~z, coming from tree diagrams
with the structure o f the one m fig 3 of sect 4
However, the lowest order contribution to the expansion o f two heavy quark
Table 1
Generahzed Wdson coefficients
Vector currents
Axial currents
Charged currents
Q2/M2
n=2
n=4
n=2
n=4
n=2
n=4
1
2
4
8
16
32
64
0 012
0 033
0 082
0 16
0 28
040
053
0 00014
0 0012
0 007
0 025
0 062
0 12
0 18
0 07
0 12
0 19
0 27
0 37
047
0 58
0 001
0 005
0 016
0 042
0 084
0 14
0 19
- 0 07
0 006
0 13
0 26
0 38
0 48
0 59
- 0 03
- 0 04
- 0 02
0 02
0 08
0 15
0 20
The number reported is the coefficient of ~-
F)
462
E Wztten/ Deepmelastwscattermg
currents in terms of the local operators of the effective theory is of order ~, coming
from one loop diagrams similar to the one in fig 4, and is proportional the the
gluon operator F~uI~JU2 Dun_ll~n
Therefore, in order to calculate the ratios R 2 and R4, we must determine the
ratio of the coefficient of the fermion operator coming from the tree diagrams to
the coefficxent of the gluon operator coming from one loop diagrams, and we must
also determine the ratio of the matnx elements of the two operators
Numerical values of the ratio of the coefficient functions are given in table 1 for
the various currents, for the zeroth and second moments o f F 2 (n = 2 and n = 4),
and for representative values of Q2/M2 A factor of ~(q2) is understood multiplying
all of the coefficients *
The numbers tend to be rather small, this is a combination of the 1IN expansion • * and phase space
Since the positlVlty of cross sections requires F2(x) to be positive, the negative
results for charged currents at small Q2 require an explanation The charged current
product ~TUp'~'TvA contains contributions representing, in the parton model sense,
the ;k quark content and'the p' quark content of the target To isolate the p' quark
content, we have to subtract from the matrix element of ~TUp'~"rvX an appropriate
multiple of the matrix element of X'yu~Tv~, which measures just the X quark content The numbers for F 2 for charged currents are thus based on not a cross section
but the difference between two cross sections, and only parton model intuition requires that this difference should be positive At small enough Q2 the intuition is
wrong The numbers for vector and axial vector charm conserving currents, however, are required to be, and are, posltwe
Also, note that the results for charged currents are based on the sum F~2 + P 2 ,
as is appropriate If one wishes the sum of the p' and ~' &strlbutlons The charmconserving currents measure this sum The difference will be considered later
Finally, note that for large Q2, the vector, axial vector, and charged current
contributions become equal, because of the asymptotic symmetry, at small Q2
there are fairly large deviations
We also need to know the ratio of the matrix element of ~tTu~Dua
Dun ~bzto
that of FaulDu2 Dun- lEarn when subtracted at/a 2 = Q2, that is, the ratio of the
upper and lower components of the two component vector
[
dt 7 * ( t ) ~ / ~ b * ~
K =- T exp~ f g*
~ - ~ ] ~ FF*,/
g.(g-(q2))
I f M 2 and Q2 are large enough, K is proportional to the eigenvector of the one-loop
* FoUowmg Gross and Wdczek (ref [2]), by ffwe mean g2/4n 2 This &ffers by a factor of
1/*r from the defimtlon used m quantum electrodynamlcs
** The manifestation of the 1IN expansion is that the group theoretical factor encountered
1
~sT(R) for a single heavy quark multlplet, or ~-,
rather than C2(G),which is equal (for
SU(N)) to N
E I~,tten/ Deepmelasttcscattermg
463
anomalous &menslon matrix corresponding to the lower elgenvalue Using the
pubhshed calculations [2] * one finds for n = 2 the result
K2--
2
'
If the gauge group is SU(3), and there are three hght quark multlplets, then C2(R ) --5-4
and T(R) = 3, and K 2 = (11/9) For n = 4 one finds K 4 = (0 126) Likely errors and
possible improvements on the use of these asymptotic formulas wall be discussed
later
The final mgredlent in obtaining numerical results is an assumption concerning
the value of ~(q2) For SU(3) the asymptotic behavior is [14]
~(q2) = 4/(9 In a2//z2) - (~2-~) (In In
a2/la2)/On a2//a2)2,
where/a is measurable m pnnclple but at present, unfortunately, unknown If, for
example, we take ~t = 0 5 GeV and M = 1 5 GeV for the heavy quark effective mass,
then at Q2 = 4/1/2 = 9 GeV 2, we obtain ~(q2) = 0 089
The final result for R 2 is &(q2) times ~ times the coefficient given m table 1
for the zeroth moment o f F 2 , for R 4 we get ~(q2) times 0 126 times the coefficient
given in table 1 for the second moment o f f 2 Under the assumptions stated m the
last paragraph, we find that at Q2 of 9 GeV2, R 2 equals 0 013, 0 03, or 0 021 for
vector, axml vector, or charged currents, whale for R 4 one gets 7 8 X 10 - 5 ,
1 8 X 10 - 4 , or - 2 2 × 10 -4 Thus we expect the heavy quark to appear at the
level of about 2% for f01dx F2(x ) but at the level of about 10 -4 for f01dx x2F2(x)
In a slmdar fastuon, one can obtain numerical estimates at other values of Q2
or with other assumptions for ~t and M or for K 2 and K 4
We would now hke to make a similar estimate of the large Q2 behavior The
result derwed m sect 5 was
JJh(q) =A (q~u,K (q2)) exp(
X
U(la,Mo,g(M))
")'NS(t)~
exp(f
~1- 3'*(t) ~[ ~ * ~
g*(~(m2)~to)
+B(q2,1z,o~(q2))~ exp( ; (M2) " 7(t) ]
* Note that, m accordance with our general result, eq (18), we want here the renormallzatlon group parameters of the theory with three, rather than four, quark multlplets, the
heavy quark is onutted
E Wttten / Deep melastw scattenng
464
X V(la,Mo,~(M)) T e x p ( f g* ~ dtT*(t)][~*]-ff~]~FF*]'
g*(-fOl )~o)
JJ(q) = C(q24z'§(q2)) T exp( g(q2)
f(M2)
at~--~)7(0
× V(p,Mo,g(M)) T exp ( S
dt ~7*(t)
/ j ' ! ~ \* F~ F * / '
(19)
g*(F(M2),Mo)
where A, B, and C are the usual Wilson coefficients and U and V are the coefficients
m the expansion of the operators of the exact theory m terms of those of the effective theory As shown in sect 5, the terms of zeroth order in ~ cancel insofar as
~ M 2) _ ~(q2) ,~ ~(M2), so we have evaluated the order ~ correctmns The only
important one-loop correction is the contribution to B 2 (recall from sect 5 that
B = (B1,B2) Is a two component vector) since the other contributions either vamsh
identically or tend to cancel m the same fashion that the leading terms do * B 2 is
simply 1/(r + 1) times the coefficient of the gluon operator m the expansmn of two
smglet currents For B 2 one obtains **
J
T(R)
B2=~a
~
4
(n+l)(n+2)
n- l
n2
n2 + n + 2 k~l ll
n(n+l)(n+2) =
(20)
For n = 2 this is -~4 8, for n = 4 It is - 0 218&
The evaluation of (19) is now simple although somewhat lengthy One needs
the quantity K discussed prewously since at the ngh~-hand side of each term there
appears the same factor as before We will, for the moment, treat the lower component o f K 2 as an unknown parameter t, but for K 4 we will use the asymptotic formula gwen previously One also needs the one-loop calculatmns of anomalous dl-
* To be precise the contributions to A and B 1 tend to c a n e d as the leading terms do The
corrections to # and ~, are hkewlse insignificant The values for B2, U, and V depend on
one's subtraction convention Our convention was d~menslonal regulanzatlon with the
understanding that the n-dtmenslonal generahzatlon of
f d4----~k
(21r) 4
is taken to be
1
( " dnk
I'(3 - n/2) n n/2-2 o (21r)4
Th~s c o n v e n t m n prevents the appearance o f extraneous terms proportmnal to Euler's
constant or the logarithm o f n With our convention, the one-loop correcUons to U and V
2
2
vamsh w h e n one sets 3/o = u, whde B2 is given m the text for Q = t~
* * This calculation had been done previously [15] I would hke to t h a n k W Caswell for
d~seusslons and for correcting an error I made
E ICttten/Deep lnelastlc scattenng
465
menslons [2] to evaluate the factors
g(M2) dt ~ )
If we let x
g(q2)
= ~(q2)/~(M2),the
final result Is
R 2 = (0 25(I - x 0 427) + ~s (t - ~-)(14 _ x0 747)
t~)/(1 + -~(t - 4)(1 - x 0 747)) ,
(21)
R 4 = 0 25(1 - x 0 02) + 0 001(1 - x 1 04) _ 0 027~
(We have dropped terms o f order ~t(1 - x ) and made various numerical approximations )
Numencal results are reported m table 2, based on the same assumptions as before for/a and M and the asymptotic value ~ for t
Although b o t h R 2 and R 4 asymptotically approach the value 0 25 as Q2 becomes mfimte, the approach to asymptopla is rather slow At Q2/M2 o f 10,000 we
find R 2 = 0.114 while R 4 = 0 0069. At a more reahstlc value o f Q2, say 40 GeV 2,
we expect the heavy quark to be present at the level o f about sxx percent m
f01dx F 2 ( x ) b u t at the level o f about 0 002 m f01dx x2F2(x )
7 2. The longttudmal structure functTon
In a similar fashion we can discuss the longitudinal structure function F L =
F2 2xF1
-
In theones m which the currents are earned b y spin-½ particles, the Wilson coefficient corresponding to F L vamshes m the tree approximation [16] It follows,
Table 2
Heavy quark contnbutlons to structure functions for large Q2
Q2/M2
foldx F2(x)
fold~.~2F2(x)
2
4
8
16
32
64
10,000
- 0 0076
0 021
0 040
- 0 0015
0 00004
0 0012
0 0022
0 0030
0 0036
0 0069
0 055
0 067
0 077
0 114
The number given Is the fractional contnbutlon of the heavy ~uark to the stated moment
of F2, under assumptions descnbed m the text For very small Q , where the operator product
expans,on is mvahd, the results are not positive defimte
E Wttten / Deep melastTcscattering
466
m an asymptotically free theory, that the moments of F L are asymptotically smaller than those o f F 2 by a factor of
To calculate F L in the large Q2 limit one needs the lowest order nonvanxslung
contribution to the Wilson coefficients These quantities have been calculated [17]
Normahzlng the coefficients so that the free field theory coefficient o f f 2 is one,
the coefficient of the fernuon operator of spin n for F L is
C2(R
n+l
The coefficient of the gluon operator is
4T(R)~
(n + 1)(n + 2)
Rather than gwe a full quantitative analysis similar to the discussion of F 2, we
wall content ourselves with consxdenng the same two hmltlng cases that we discussed m connection with (15)
If Q2 is large compared to M 2 but the difference between ~(Q2) and &(M2)
can be ignored, then the fermxon SU(3) slnglet operator does not appear in the
expansion of a heavy quark current product However, the gluon operator appears
in this expansion with the same strength as m an expansion of a hght quark current
product Considering the special case n = 2 and using C2(R ) = 4, T(R) = 2 for the
four.quark tnplet theory, and the asymptotic value ~ for the ratio of the gluon
operator to quark operator matrtx elements, one f'mds that the heavy quark contribution to f01 dx F L xs ~ of that of a single hght quark For n = 4, corresponding
to f01dx X2FL, the heavy quark contnbutlon is only 11% that of one hght quark
In the mfimte Q2 limit, the comments following (15) are vahd, and the heavy
quark contnbunon to e F L uals that of a hght quark
When Q2 is of order 4M 2 the above conslderatmns are not vahd and one must
calculate the generahzed Wdson coefficients m the sense of sect 6 Rather than
reporting detailed results, we will only state the following for Q2 = 4M 2 for vector
currents, the heavy quark fractional contnbunon to f01dx E L IS strongly suppressed (by a factor of about 10) relative to the value ~ , for charged currents the
suppression is roughly a factor of 2, and for axml currents it IS about 30% Actually
for charged currents there is for Q2 ~ M 2 an extra contnbutlon to F L proportional
to the k quark operator, since the tree dmgram no longer vamshes The extra contributlon is calculable in terms of the X quark distribution and might be measureable at Q2 of a few GeV 2 and rather small x m AC = AS = 1 transitions
7 3 The panty vtolatmg structure functton
The consideration o f f 3 is rather simple the charmed quark makes (nearly) zero
contribution to F 3
F 3 arises from an Interference between a vector and an axial vector current and
E ICttten / Deep melastzcscattering
467
so is odd under charge conjugation However, the gluon operators of twist two are
all even under charge conjugation Therefore, the gluon operators do not contribute
to F s [181
In fact, the only operators relevant to F 3 are the fermlon operators of odd spin
These are multlphcatively renormallzable without mixing with the gluon operators
In our "small Q2,, discussion, the entire calculation consisted of evaluating the
coefficient of the gluon operator at the one-loop level Since this now vanishes, the
heavy quark contribution is suppressed by an extra factor of ~, and so we expect
it to be neghglbly small
At large Q2, we reason as follows Since the mixing between fermlon and gluon
operators is now absent, the cancellation between leading terms in (15) is now not
approximate but exact. As can be seen from (15) the asymptotic value, at very
large Q2, of the heavy quark contribution is now not 0 25 but rather is a constant
determined by the difference between the smglet and the non-smglet anomalous
dimensions and also the coefficients U and V Since the order ~ contributions to
these quantmes vanish, the constant limit of the heavy quark contribution to F 3
will be of order (~(M2)) 2 It will also be suppressed by the explicit factor of ¼ appearing m the free field theory values of B and U in (15), by a factor of 1/N from
the large N expansion, and by a factor of 1In 2 for large spin n Thus, although we
obtain a scaling result for the heavy quark contribution to/73, we expect it to be
negligible in magnitude
Note that the smallness of the heavy quark contribution to F 3 does not follow
from a symmetry argument alone Symmetry ensured that the leading term vanished, but the corrections are non-zero and are small only because of asymptotic
freedom
7 4 Charmed quarks and charmed anttquarks
In a like fashion, we can show that in an asymptotically free theory the dlstrlbuhen of heavy quarks is equal to the distribution of heavy antlquarks
In the parton model sense, ~ receives a contribution from the p' content of
the target while/~2 receives a contribution from the if' content No other structure
functions distmgmsh p' from ~', except F 3, to which the heavy quark does not
contribute To show that the heavy quark and antlquark dlstnbutlons are equal is
thus a question of showing that the heavy quark does not contribute to F~2- ~- for
any target
P2 is determined by the matrix element of
f d x e'q X(T(V(x) V+(0))+ T(A(x)A+(O))),
where V and A are the charge raising vector and axial vector currents ~2 Is determined by the matrix element of
fox e'q X(T(V*(x) V(O))+ T(A+(x)A(O)))
E I1htten/Deep inelastic scattering
468
Charge conjugation interchanges these objects, so that the difference between them
is odd under C Thus the previous &scussxon showing that the heavy quark does
not contribute (appreciably) to F 3 also shows that It does not contribute to
F~2-v
75 x dependence of the structure funcnons
The quantities that we have discussed so far are the moments of the structure
functions Actually, we can combine the moment relations to obtain very severe
constraints on the x dependence of the structure functions In particular, we wdl
see that heavy quarks wdl be found almost exclusively at very small x
For example, let us first consader the charmed quark contribution to vector
current structure functions at Q2 = 4M 2 = 9 GeV2 In subsect 7 1 above we denved the following sum rules, whxch we now state in parton model language
1
1
f
+/3')--oo13
f
0
0
I
1
f d x x 2 ( p ' + / 3 ' ) = 7 8 X 10 -5 f dxx2(q + 0 ) ,
(22)
0
0
where q and ~ are the total quark and antlquark distributions From existing experimental data [4]
1
1
f dx x2(q + q) "~ 0 1 f
+rT)
0
0
Combining these relations, we conclude that
1
(23)
1
f dxx2(p ' +/3')= 6 X 10 -4 f dx(p' +/3')
(24)
o
o
This estimate could be in error by a factor of two, but hopefully not by a factor
of ten
Eq (24) lmphes that the contribution to f01dx(p ' +/3') is concentrated at extremely small x In fact, one easily deduces that
1
1
f dx(p' +/3') ~< 0 06 f dx(p' +/3'),
01
0
1
1
f dx(p' + if') ~< 0 24 f dx(p' +/3')
0 05
0
Thus, at Q2 = 9 GeV2, the first moment of the charmed quark distribution is at
least 94% saturated by the region x ~<0 1 and 76% saturated by the region x < 0 05
Hence at this value of Q2 the charmed quark contnbutxon to f015 dx(q + ~)
will be less than (0 24)(0 013) or about 0 003 of the total, while the charmed quark
E Witten / Deep melastw scattenng
469
contribution to f l ! dx(q + F/) will be less than (0 06)(0.013) or about 0 0008 of
the total And these are only upper bounds ) The actual values may be much less
Thus, efforts to detect a charmed component of the nucleon at Q2 = 4M'2 and x
greater than about 0.05 or at most about 0 1 are bound to yield a null result
What is the sltuatmn f o r x ~< 0 059 So far we know that at Q2 of 9 GeV 2,
0 O5
001 ~<
dx(p' + p')
f
0
~<0013
1
f dx(q + ~ )
0
Experimentally,
o 05
f
~(q + 0)
0
1
f ax(q+q)
0
IS roughly one tenth Hence
0 05
0 10~<
f
0
ax(p' + p ' )
<~0 13
0 05
f
0
(25)
dx(q+ q)
Thus, charm is actually a ten to thirteen percent effect in the region o f x less than
0 05.
At larger values of Q2, the charmed quark may appear at larger values o f x At
Q2 of 50 GeV 2 (or about 22 M 2) one finds (based on table 2 and the experimental
result (23)) that f01dx(p ' + if') is almost 90% saturated by the regmn x ~< 0 2 The
charmed quark ls approximately a 6% contribution to f01dx(q + t~) but wdl contribute at the level of about ten percent to fo 0 2dx(q + ~ ) and at the level of at
most about t% to fo 1 dx(q + 0")
At the other extreme, at Q2 = M 2 = 2 25 GeV 2, we find that 3 of the charmed
quark distribution ls at x less than 0 02 The charmed quark wall contribute to
f o l ~ d x ( q + FI) at the level of at most about one tenth of a percent but to
f8Dr92 dx(q + ~) at the level of five percent
So far we have only derived upper bounds on the maxamum value o f x at which
charmed quarks make a large contribution To derive a lower bound, one may assume that at no value o f x is the charmed contribution to (q + q ) greater than 25%,
E I¢~tten / Deep melastw scattermg
470
and that it is a monotomcally decreasing function o f x If the charmed component
is exactly 25% for x less than some crmcal value and zero for x larger, then from
(25), at Q2 = 9 GeV 2, the critical value must be approximately 0 02 or 0 025 Tlus
can be taken as a minimum value o f x below which charmed quarks and antlquarks
are certainly a large contribution At Q2 of 50 GeV 2 or 2 25 GeV 2 the correspondlng minimum estimate for x is 0 08 or 0 004
What lepton beam energies are reqmred in order to detect the charmed component of the nucleon9 Since the beam energy is, for small x, E = Q2/2Mx, one would
need a beam energy of at least 56 GeV to reach x of 0.02 at Q2 of 2 25 GeV 2,
90 GeV to reachx of 0 05 at Q2 of 9 GeV 2, or 125 GeV to reach x of 0 2 at Q2
of 50 GeV 2 These are the minimum beam energies at which one m~ght hope to
see an effect, and a beam energy several hmes larger may be required, especmlly
tf the minimum estimate for x ~s more reahstlc than the maximum one
8. Sum rules
Unfortunately, only certam special current products are measurable in practice
To test our results experimentally, it IS necessary to derive sum rules relating the
structure functions for different processes We will here state briefly the sum rules
that could be used, m principle, to measure the charmed quark content of the
nucleon
For slmphclty we will assume the vahdlty of the popular four quark triplet
model [3], which contains a smgle unknown parameter, sm20w Moreover, we
will formulate sum rules only for the asymptotic reglon Q2 >> 4/142. For Q2 ~ 4M2
the predictions are more complicated because the heavy quark contribution depends
on which current one considers, but our discussion could easdy be extended to
account for this if experimental results become available
We will find it sufficient to consider only processes with lSOSCalar targets To
measure the sum (p' + if'), the relevant structure funchons are, in parton model
language, the following [19]
For electron or muon scattenng,
F~ m = x ( ~ (p + i f + n + ~ ) +-~(),+ X-) + ~ ( p ' +/~'))
For charged currents one can consider e~ther the total cross secUon or the cross
section for AC = AS = 1 processes only * For the average of v and ~-events, one gets
~+b-(total) = x((p + p + n + ~) + (~ + ~) + (p' + if')),
~ + r ( z x c = ~ s = 1) = x ( ( x + ~) + (t,' + p ' ) ) cos20 c
*Modtficattons of our statements based on the total C = 1 cross section (without dlstlngmshmg strangeness changing from strangeness conserving events) are also possible
E ICltten /Deep melastlc scattering
471
The neutral current structure function is
F~ eutral = x((½ - sin20 w + ~ sln40w)(p +/~ + n + ~-)
+ ((~ - -~ sln20w)2 + ¼)(X + ~ ) + ((~ - ~ sln20w)2 + ¼)(p' + f i ' ) )
These four structure functions are determined by the three unknowns
(p + i~ + n + ~), (~ + ~-), and ( p ' + io'), so that an accurate measurement of any three
of the four would permit one to extract ( p ' + iO')
One would use the same combinations of structure functions to study F L
It is similarly possible to derive sum rules mcorporatlng the predlctaon that the
heavy quark will not contribute to F 3 or to F~2- ~The most reahstlc posslblhty revolves charged current data for-charm-changing
transitions
~-~-(Ac
= A S = 1) = x ( ( X - ~-) - ( p ' - ~ ' ) ) c o s 2 0 c ,
xF~3+ff(AC = AS = 1) = -x((;k - X-) + (p' - / ~ ' ) ) cos20 c
Thus the relation F~2-~-+ xF~3+~-= 0 for AC = AS = 1 processes expresses the fact
that p ' = iO'
If it is possible to measure F 3 from neutral currents, an ad&tlonal sum rule IS
possible For lsoscalar targets
xF~3eutral = x((½ - s l n 2 0 w ) ( - p + fi - n + h-)
+ (~ - ] s l n 2 0 w ) ( - h + X-) + (½ - -} s l n 2 0 w ) ( - p ' +/~'))
This structure function contams a new term, ( - p + ~ - n + if), which is rodependently measurable, in principle
x~+~-(AC = AS = 0) = x ( - p + p - n + n ) cos20 C
Thus, one obtains an additional relation among the above four quantities
What can we learn about the sum rules from existing data 9 The existing data,
based on measurements o f F ~ m and/~2 +~- (total cross section) permit the statement
that f01dx(p ' +/~') is less than 0 06 of the total quark distribution [20], the experimental result is actually consistent with zero The data come mostly from Q2 ~<
5 GeV 2 A look at table 1 for Q2 = 2M 2 (4 5 GeV 2) for electromagnetic (vector)
currents shows that we expect f01dx(p ' + if') to be about 0 005 of the total, while
for charged currents we expect an even smaller contnbutlon Thus, the existing
data do not provide a real test
9. Error estimates
Finally, it remains for us to attempt to estlmate the error that we expect m our
predictions We wall consider m turn the major sources of uncertainty
472
E I4htten/ Deep inelasticscattenng
9 1 Mass correctzons
We have consistently assumed that terms of order I[M2 can be ignored Since
/1/2, at 2 25 GeV 2, is not much larger than the nucleon mass, tlus procedure is
dangerous, and it is possible that, on this account, our calculation is ~rrelevant to
the actual charmed quark.
On the other hand, precocious Bjorken scaling may lead one to suspect that the
relevant mass scale is rather small In a study of two-dimensional Yang-Mflls theory,
Callan, Coote, and Gross [21 ] found that it is quark masses that set the scale for
corrections to the leading short distance behawor This suggests that the 1/M2 terms
may actually be of order m2/M2, where m is a light quark mass In this case, we
would expect them to be negligible, especmlly since they would be suppressed by
factors of the sort that cause the leading terms to be small (factors of 6, of l IN,
and phase space factors that enter m the calculation of short distance coefficient
functions).
One aspect of the mass corrections problem, the proper choice of scaling variable,
is less senous here Our results are relatwely stable against a change in scaling vanable, since we are predicting the actual magnitude, and not only the Q2 dependence,
of certain combinations of structure functions
9 2 The effectwe couphng constant
In makang numerical estimates we have made a rather optimistic assumption
concerning the effective couphng constant We took/a = 0 5 GeV in the asymptotic
formulas, which corresponds to ~(M 2) = 0 14 This Is consistent with estimates
based on charmomum models [22], but data for e+e - annihilation just below the
new resonances suggest a larger value As long as ~ M 2) Is small anough as to be a
reasonable expansion parameter, It would, of course, be simple to revise our estimates when a determination of 6 becomes available
9 3 Coefficients of non-leading operators
An additional source of error and, except for n = 2, perhaps the most serious,
arises from our use of the asymptotic formulas for the ratio of the matrix elements
of the gluon and fermlon operators o f twist two
The correction to the asymptotic formula vamshes hke a power of 6 , the power
IS determined by the difference between the two elgenvalues o f the anomalous dimension matrix of the smglet operators For n = 2 the power Is 6 2/3, for n = 4 it is
1 04 Thus the ratio of the gluon to fermlon matrix elements is actually
+ O(ff(M2) 2/3) for n = 2 and 0 126 + O(~(M2) 1 04) for n = 4
Smce 6 ( M 2) is hkely to be of the same order of magnitude as 0 126, this Introduces an uncertainty of perhaps a factor of two (or more) m the prediction for
n = 4, an uncertainty which we have no way to control
E Witten/Deep melasttc scattering
473
This uncertainty becomes more severe for larger n, which is why we consider
only sum rules for n = 2 and n = 4
However, for n = 2 it Is possible to experimentally determine the ratio of the
gluon to quark matrix elements [2] Tlus is so because the sum of the gluon and
quark matrix elements is the known matrix element of the energy momentum
tensor, while the quark matrix element can be measured by determlng z =
foldx(q + ~ ) The best existing measurements are z = 0 51 + 0.08 from electron
scattering and z = 0 51 -+ 0.05 from neutrino scattering The ratio of the gluon to
quark matrix elements is (1 - z)/z, so if one accepts the value z = 0 51, one estimates that the ratio of the gluon to quark matrix elements is 0 98, in contrast to
the asymptotic value ~ In the "small Q2,, region, this merely reduces our estimate
~::~;hreeghar?oede qmUa:kCetttnb0u;8oT~f~ 1.O~Cn(qfin~dI bYa~ffa:t~ of ~he Irnnthee~rSYm
mp"
50 to 100 GeV 2, our previous estimates are reduced by about 30%
10. Conclusion
The picture of the charmed quark content of the nucleon that emerges in asymptotically free non-Abehan gauge theories IS rather appeahng The smallness of the
charmed quark distribution and the peaking at small x are what one would expect based
on experience with X quarks and also with antlquarks The absence of a p' contribution to F 3 is consistent with the fact that F 3 seems to vanish at small x, where the
p' quarks are concentrated The equality of the p' and if' distributions corresponds
to the fact that the heavy quarks are not genuinely participating in the dynamics
of the light quark state, so that the theory is not sensitive to the fact that p' and if'
carry opposite non-Abehan charge
Our results are in reasonable agreement with recent work by Goldman [6]
Based on a Drell-Yan calculation of ~k photoproductlon, Goldman placed very
severe constraints on the possible charmed component of the nucleon We agree
with Goldman that in the small energy region in which measurements of the quarkantlquark sea now exist, there will be a large suppression of charm relative to the
rest of the sea We are unable to make a detailed comparison with this work, because our result Is not really a parton model result, we predict a rather large Q2
dependence of the charmed quark distribution
An experimental evaluation of our sum rules would require measurements accurate to a few percent with lepton beam energies of the order of 100 GeV Such
an evaluation, if it became possible, would provide an interesting, although rough,
test of the idea that the strong interactions should be described m terms of a nonAbehan gauge theory of quarks and gluons
I would like to acknowledge many valuable suggestions by D.J Gross throughout the development of this work I would also like to thank W Caswell for several
discussions
E Wltten /Deep melastzc scattermg
474
Appendix
We would like to explain more precisely how one obtains the generahzed Wilson
coefficients for the expansion of a heavy quark current product These coefficients
were important m the phenomeno]oglcal discussion, some of them are given in
table 1
The utdlty of the expansion is that the coefficients, together with the math×
elements of the twist two operators, determine the moments of deep inelastic
structure functions The simplest method to obtain the coeffioents is, accordingly,
to choose a convenient target, calculate the relevant moments of structure functions, and dwlde by the matrix elements of the twist two operators
For our purposes, far the most convenient target state to consider is a gluon
The matrix elements of the relevant twist two operators are all equal to one to
lowest order, so one need only calculate the structure functions for current-gluon
scattering at the one-loop level These are obtained simply by calculating the total
cross section for current + gluon -~ quark + antlquark in lowest order
For purposes of reference we give here the various structure functxons F~rst we
consider charm conserving vector and axial vector currents
Let o = x/1 - 4M2/s be the center-of-mass frame velocity of one of the quarks
Then for vector currents we obtain
F V = ~ u x [ - 1 + 8x(1 - x) - 4M2x(1 - x)/Q 2 ]
+ ½ x [1 - 2x(1 - x ) + 4M2(x - 3x2)/Q 2 - 8x2M4/Q 4] ln((1 + o)2s/4M2),
F v = 2x2(1 - x ) o - 4(M2/Q 2) x 3 ln((1 + o)2s/4M 2)
For the difference between axial and vector currents one obtains
F A-V = 2(M2/Q 2) x ln((1 + o)2s/4M2),
F t - V = -4(M2/Q z) x2(l - x) v + 2(M2/Q 2)
X (x + 2x2(1 - x(l + 2M2/Q2))) ln((1 + o)2s/4M 2)
The above structure functions are all understood to contain a threshold factor
O(s - 4M2), and a factor ofg2/47r 2 IS also understood
For charged currents, the structure functions are as follows
F~ h =(1 -M2/s)x2(1 - x ) ( 1 + M2/Q 2)
-'-~+3x2(1-x)
1--~
1
(l
--~-~
)
M2
+ 4 (1 + M2/Q 2) [1 - 2x(1 + M2/Q2)(1 - x(1 + M2/Q2))]
E tCltten/Deep melastlc scattenng
475
M2 (1 - 2x) ln(s/M 2)
X (ln(s/M 2) + ln((s - M2)/m2)) + 3x 2 ~-~
F~ h =(1 - M 2 / s ) ( 2 x 2 ) ( 1 - x )
(1
M2
M2
+-ix M~-~2 [1 - Z~(1 +M2/Q2)O - x ( l +~U~/Q2))I
X (ln(s/M 2) + ln((s - M2)/m2))) + 2x2(1 - 2x)
ln(s/M 2)
The threshold factor can now be taken as O(s - M 2) The quantity m refers to
the mass of the light quark, which one sets equal to zero except where this would
yield an infrared divergence.
For charm conserving currents the generalized Wilson coefficients are simply
the moments of the above structure functions For charged currents, there is, as
discussed in subsect 7 1, an extra subtlety, the moments of charged current structure functions are sensitive to the light as well as the heavy quark content of the
state To isolate the heavy quark contribution one must subtract from each moment o f the charged current structure functions an appropriate multiple of the
corresponding moment of a light quark current structure function The scale of
the subtraction can be discovered by studying the generalized Wilson coefficmnts
at the tree level or simply by observing that the infrared divergent term proportional to I n m 2 must cancel
References
[1] R P Feynman, Photon-hadron interactions (Benjamin, New York, 1972)
[2] D J Gross and F Wflczek, Phys Rev D8 (1973) 3633, D9 (1974) 980,
H. Georgl and H D Pohtzer, Phys Rev D9 (1974) 416
[3] S Glashow, J Ihopoulos and L Matanl, Phys Rev D2 (1970) 1285,
S Wemberg, Phys Rev Letters 19 (1967) 1264
[4] D C Cundy, Neutnno interactions, Proc 16th Int Conf on high-energy physics, London,
1974 (Rutherford Laboratories, Chflton, Dldcot, Berkshire, England, 1974) p IV-131
[5] G 't Hooft, Nucl Phys B72 (1974) 461
[6] T Goldman, Upper bound on charm quarks m nucleon from the cross section for ~(3100)
photoproductlon, SLAC prepnnt (April, 1975)
[7] T Appelqulst and J Carrazone, Phys Rev Dll (1975) 2856
[8] S Wemberg, Phys Rev D8 (1973)3497,
G 't Hooft and T Veltman, Nucl Phys B44 (1972) 189
[9] C G Callan, Jr, Phys Rev D2 (1970) 1541
[10] S Joglekar and B Lee, General theory of the renormahzatlon of gauge mvanant operators,
NAL preprmt (September, 1975)
[11] S Wemberg, Phys Rev 118 (1960) 838
476
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
E Witten / Deep melastw scattering
C G Callan, Jr, Phys. Rev D5 (1972) 3202
N Christ, B Hasslacher and A Mueller, Phys Rev. D6 (1972) 3543
W. Caswell, Phys Rev Letters 33 (1974) 244
W Caswell and R Schrock, private commumcatlon
C G Callan, Jr and D J Gross, Phys Rev Letters 22 (1969) 156
A Zee, F Wdczek and S B Trelman, Phys Rev D10 (1974) 2881
A De Rujula, H Georgland H D Pohtzer, Phys Rev D10 (1974) 2141
G Altarelh, N Cabblbo, L. Malam and R Petronzlo, Phys Letters 48B (1974) 435
M K GatUatd, Search for charmed parUcles, Fourth Int Conf on neutnno physics and
astrophystcs, Downmgton, Pa, 1974 (New York, 1974) 65
[21] C G Callan, Jr, N Coote and D J Gross, Two-dimensional Yang-Mfllstheory, a model
of quark confinement, tMneeton preprmt (September, 1975)
[22] T Appelqmst and H D Pohtzer, Phys Rev Letters 34 (1975) 43