How strong are strong interactions?

Nuclear Physics B76 (1974) 365-374 North-Holland Pubhshmg Company

HOW STRONG ARE STRONG INTERACTIONS?* Harald FRITZSCH ~" and Peter MINKOWSKI

CaliforniaInstitute of Technology, Pasadena,Cahforma 91109 Received 11 March 1974 (Revised 9 April 1974) Abstract It is suggested that strong interactions are described by the renormahzable, asymptotically free gauge field theory of colored quarks and gluons and that the renormalized quarkgluon coupling constant is small, when evaluated at momenta of the order of the proton mass The true asymptotic predictions of this theory for deep inelastic scattering are shown to be irrelevant for present experiments Scalingis found to be violated logarithmically according to lowest order perturbation theory Upper bounds for the strong interaction coupling constant are derived In certain situations (e g , deep inelastic scattenng) the electromagnetic and weak currents of hadrons seem to act as in free quark theory The understanding of this behavlour from a field theoretic point of view Is still Incomplete Recently some progress has been made Using the renormahzatlon group technique [ 1 , 2 ] , it was shown that in non-Abehan gauge field theories such as the one in which colored quarks are coupled to color octet vector gluons it can happen that the effective coupling constant approaches zero logarithmically at very high energies and therefore canonical Bjorken scaling is violated only by specified powers of log q2 [3]. The hght-cone algebra of currents in this model deviates from the free quark hghtcone "algebra in two respects (l) In the SU(3) smglet channel appear besides the quark bdmears . %

Q(X)Tul

VgL2 run -~ q(x)

also the gluon bdmears

A

GA

.

G o

/'tlOV/'L2 ~7~Un-1 A ~ n

'

A the gluon field strength, A the color Index, Vu denotes the covarlant derivative, Guo A=I 8 * Work supported m part by the U S Atomic Energy Commission Prepared under Contract AT (11-1)-68 for the San l:ranclsco Operations Office, U S. Atomic Energy Commission I" Work supported in part by Deutsche Forschungsgememschaft

366

H Fntzsch, P Mmkowskt, Strong mteracnons

(n) There anse anomalous logarithmic smgularmes depending on the couphng constant g (S~/n(x 2, g), SG(x 2, g)) muhlphed by the corresponding quark or gluon bllinear respectively In the hmlt g ~ 0 one recovers, of course, the free quark result

sq(x 2, O) = l,

S(n](x2 , O) = 0

(l)

In the Bjorken hmlt one has typically the behavior f d ~ ~"-2F2(~, q2) ~

bn [log(_q2)l-an E ,

n = 1,

o%

(2)

where/~n are the hadron expectauon values of tile spin n tensor operators, the an are calculable numbers and the b n are unknown constants The behavior (2) IS compatible with present experiments, although rather discouraging using (2) it IS impossible to deduce from the present expernnental data the dynamically interesting expectation values E n, which are related to the quark and gluon dlStrlbuUons inside the hadrons Present experiments would have nothing to say about them In this note we should hke to point out that the behavior (2) as predicted by the asymptotically free quark-gluon field theory, may well be totally Irrelevant for present experiments Perhaps it is even irrelevant for any foreseeable leptoproductlon experiment, "although it may be the ultimate asymptotic behavior of the moments of the structure functions m the Bjorken hmlt We assume that the strong lnteracuons are described by a canomcal Yang-Mllls gauge theory revolving colored quarks and gluons The color gauge group is fixed to SU(3) (color), provided the interpretation of the n 0 decay IS right [4] In order to accommodate exotic posslblhtles like charm, we take the group SU(m) × SU(m) as chlral group (m = 3 for ordinary quarks), i e , we have the group SU(m) × SU(m) X SU(3) (c°l°r) We further assume that the renormahzed quark-gluon coupling constant ~s small when evaluated at the q2 value.ta2, at which Bjorken scahng sets in K = g 2 / 4 n < 1 To be defimte we take/a 2 = m 2 ~ 1 GeV 2 At first ~t seems to be absurd to assume that the strong mteractlons are described by a field theory with small couphng However, one should keep m mind that strong interaction verhces like the pxon-nucleon vertex describe the coupling of comphcated bound states of quarks and gluons From them very little is learned about the strength of the basic quark-gluon vertex Below we shall consider two distinct cases which differ wnh regard to their lmphcations for experiments Case A K not very small ~ < K < t~ Case B ~ very small K < ~ (for example ~ = od) We introduce a scale parameter X = (-q2//.t2)½ and define t = log h = ½log(---q2/la 2) The renormahzauon group equaUon for the effectwe coupling ~(t, ~) = ,~2(t, g)/4n then takes the form [5]

tl Frttzsch, P Mmkowskt, Strong mteracttons

367

_d g(t, K) = -2B g 2 + O(g 3), dt

(3)

where g(0, ~) = K and B = 1/47r(11 ~m)(=9/4rr fl)r ordinary quarks) Since K is assumed to be small we can apply perturbation theory to (3) and find to lowest order ~(t,K)_ K

1

"-

(4)

i+~Bk-t

The solution (4) describes the behavior of the effective coupling constant as long as ~¢ is small compared to one, in particular in the region 0 < t < <~(or _q2 > / j 2 ) The "anomalous dimensions" of the SU(m) non-smglet tensors of spin n (quark blhnears) are independent of m and given by

(5)

Tn(K) = CnU + 00¢2),

where [5]

4O n(n+l) :

Cn

27r 3

(C 1 = 0 ,

C2

+4

_ 1 32 21r 9 '

A'l ~-

1 16 Iogn)

Cnn._,o. >27r 3

(6)

"

In the SU(m) sxnglet channel mixing of quark and gluon blhnears occurs in higher order perturbation theory We consider especially the case n = 2, where the quark and gluon parts 0 q , 0 G ~ of the energy-momentum tensor appear The matrix 72(•) can be dlagonahzed using the linear combinations

Our = oqtav + OGuv + trace terms,

Ruv

=

0quv

-

~

(7)

0 uv G '

(8)

in case of ordinary quarks (m = 3) [6] The energy-momentum tensor has, of course, no anomalous dimension ( ~ = 0), but for Ruv one finds _150 7R - 2~ -9- K + O(• 2)

(9)

Applying standard renormallzatlon group techniques we obtain for the Fourier transforms of the "anomalous" singularities the following representations t

:

t

: (1 +

,

(10)

368

H Frltzsch, P Mlnkowskt, Strong interactions

where ° s ~ (K = o) o. =~K-

In the ultraviolet hmlt one has, of course,

q2 S~(:-'~!,.2I'"t-+(u.B) -Cn/2B [ l °. g2- q]2

- Cn/2B ,

(ll)

which Is a specml form o f ( 2 ) where the constants b n are now gwen by b n = (KB)-Cn/2B The behavior (11) Is expected to set m for 2BKt = Br log

__q2/p2 >~ 1, hence

_q2 >>el~Be /a2,

for (12)

which means for ordinary quarks m CaseA

_ q 2 > > 5 0 G e V 2,

CaseB

~q2>> 106GeV 2.

(13)

(IfK = ix, one must have ._q2>> 1080 GeV2)) Hence in any case the logarithmic behavior (1) Is hkely to be irrelevant for energies explored so far It would be irrelevant for any feasible expertment if ~ is very small, e g , K ~ tx We interpret the observed scahng behavior m the deep melasuc region as follows In the energy region tested so far only the 1-term m the expansion (10) Is seen (Born term), which is gwen by free field hght-cone algebra or by the parton model At higher energms we expect modifications of the nawe scahng behawor For SU(3) octet channels one finds

[

f F~ctet(LqZ)~n-2d~

o

' E°ctet(l+-gOn ) I+BK

-q~=

log{-q2]-cn/

(14)

~,-~lJ

Since B~ log-q2//a 2 < 1 m the present energy region, we can expand (14) and find the corrections to the canonical scahng behawor as gwen by second order perturbatlon theory*

fF~ctet(~,q2) ~n- 2d~j _-ln-OCtet.l,q2-) ~_ Eoctetn

1-~: o n + ½C n log

0 In particular we have * Scahng vmlatlons of this sort were first discussed m 17]

(15)

H Frttzsch, P MmkowskL Strong mteracttons

369

~ctet(q2) _ .octet. ,2. Eoctet ,2 tn (q ) " n ½KCnlogq~-, qZ n=2 12

( q_2._ )

"r. 2

t~ L '~rr log q2

(16)

We should hke to point out that the Ioganthmm corrections to the scahng laws of the type ( 15, 16) are independent of B and m That means m particular, that they are mdependent of the fact that the theory ~s asymptotlc',dly free To test the sign and magmtude of B, at least correctmns to scaling of the order K2 have to be measured We discuss the mixing of quark and gluon bdmears for the interesting case n = 2, m = 3 The SU(m) smglet channel is projected out by the hnear combmatlon

/l

4 ( q 2) = ~ 6 fd~(/,~P 0

1

+/~,~n) _.. fd~(f~ + f ~ n)

/

,

(17)

0

which we normahzed such that J~2 = 1 for free quarks The relevant tensor operators are Our and R ~ (see (7, 8)). To zeroth order m ~ only the quark part of the energymomentum tensor 0q~, appears m the current product expansmns which for ordinary quarks can be decomposed as follows 0 qI~V ='29gOlav + , ~ R ~zv

(18)

Hence the asymptonc behavior of (17) IS (19) Here one has used the fact that ~pl0~lp) = 2pup v E2(R ) Is the proton matrix element of R ~

( p l R ~ IP) = 2E2(R)pup v + trace terms

(20)

It IS gwen by the quark part of the total hadron momentum, defined by (pbOq~lp) = 2epupv + trace terms E2(R)=~

e-~

(21)

In the true asymptotm region (~ ~ 0) one finds J~2(q 2) ~ 9 relevant for present energies, one obtains of course 4 ~ e,

To zeroth order, (22)

and the first order corrections to scahng are given by _8 4 ( q 2 ) - 4 ( q ' 2 ) "" E2(R) 9n

q,2 _ 2 5 e - 9 ,2 log q2 187r K log ~

q~

(23)

370

H Frttzsch, P Mtnkowskt, Strong mteracttons

Taking the data of the neutrino production experiments seriously one finds that about 5(Y); of the proton momentum is due to quark degrees of freedom (e ~ ½) [81 which gwes ,2 jO(q2) _ jOg(q,2) _~ 0 06 K log-q-a(24) -

-

qZ

The consequences for electroproductlon can be estimated as follows Using the experimental data [9] 1

I~P=

f dI~F~P = 0 176 -+ 5%,

0 I ~n =

1

f0 d~F~n = 0 142 +- 5%,

(25)

and information about e, one can estimate the amount of smglet and octet contribution to the electroproducuon scahng functions ~e = ~ ' u + ~)"d + ~)"~ = ~ ( ~ + i~d+ )"~)+ -~(I'~ + ~d I - f f ~ ) + t6(I'~--~'d), (26) I2 2 where )"~ denotes the integral over the corresponding quark and antNuark dxstnbuuon funcUon 1 I"~ = f d~ ~(q (~) + F/(~)) 0

(27)

The first order corrections to scahng are then gwen by ~2p(q2 ) _ l~P(q,2)... ~(4(q2) _ 4 ( q ' 2 ) ) +(7"7 _ De) 8 t ¢ log q,2 ,2 1)K logq~ -

(28)

e ~ 0 5 and (25) one finds ,2 A~2P ~ 0 03 K log ~

(29)

= ~1( e

+ 87~ p

--

Using

Since no wolatxon of scahng ~s seen so far m the electroproductmn experiments, z2J2/l 2 has to be smaller than the error of 5%. Therefore we conclude

o g/log

5

(q2max~ /

The experimental data show that [~P stays constant w~thm the q2 range of 15 GeV 2 [9], hence we have

(30)

H Frttzsch, P Mmkowskt, Strong mteracttons

K<~

371

(31)

Besides tile logarithmic corrections to the scahng laws there are "also powerhke correctxons of the type M2/( - q2) where M 2 is a measure for the effecuve strength of the trace of the energy-momentum tensor Assuming that power-like corrections to scahng, wluch are of course present for low q2 values, can be transformed away by using &fferent variables hke ~' = 1/of = ~q2/2u+m 2 , one may lower -q2mm to 1 GeV 2 or less, m which case one finds a strong restriction from (30)

K ~< ~

(32)

If(32) were really true, It would be almost impossible to see in the experiments the asymptotic behavior as predicted by the asymptotically free theory, one would need ._q2 ~ 105 GeV2T Evidently the electron and neutrino production experiments at NAL, for which q2max wdl be at least 50 GeV 2, are able to determine K ff one is m case A, or estabhsh case B Local currents have no " anomalous dimensions" (71 = 0) Hence one obtains from (10) for example, the sum rule* 1

½fd~(F~P(Lq2) + F~n(~, q2)) ~ ...-3(i + k O l ) + O( M[} ,q-, 0 "" ~-3 (1 + ~:o I +

) + OIM-I

\q2 ]'

(33)

where o I is of order 1 In the true asymptotic hmR the right-hand side o f ( 3 3 ) is, of course, - 3 , but for log Iq2//a21 '~ I/B~ (present energy regxon) we find a shift hnear m K, to this order there are no logarithmic corrections and the sum rule IS approached power-hke For e+e - annthdatton into hadrons we obtain m the case of ordinary quarks

Oe+e-* ~+~I -

winch behaves in the true asymptotic region hke [I I] R ( q 2 ) ~ 2(1 ÷

4/9 ) logq2//a 2

However, at present energies we expect R(q2) --* 2(1 + ~ ) + O(K 2)

(35)

* In the true asymptotic hmlt the sum rule (33) is the sume rule discussed first by Gross and Llewellyn Smith [ 10]

372

tl Frttzsch, P Mmkowskt, Strong mteracttons

Needless to say that formula (35) is not able to explain the increasing cross section ratio seen in the experiments Note that the shift 2K/rr arising to first order m • xs less than one m any case It would be lnterestmg to measure the logarlthmxc corrections to scahng as described above They are the strong interaction analogue of second order effects m quantum electrodynamlcs like the anomalous magnetic moment of the electron. l f r is much smaller than ~ , for example of the order of a, there is no hope to see any logarithmic deviations from scahng. In case ~ ~ a one fmds ,2 l~P(q2) _ l~P(q,2) ~_ 2 " 10 4 log ~ (36) It is unreahstlc to beheve that at energies as large as required to test (36) the whole approach Is stall jusufied First of all it would be necessary to include also corrections from electromagnetic and weak interactions. But also gravitational effects become important, since one ~s testing distances of the order of Plancks elementary length (10- 33 cm) Thus, if the basic quark-gluon couphng constant ~s indeed of the order o f a ~t is sufficient for phenomenologlcal purposes to assume that current commutators behave near the hght cone as m free quark theory. The free quark hght cone singularities are vahd up to distances o f the order of Plancks elementary length Of course, we do not dare to speculate what happens there. If, as we have assumed throughout, the basic field theoretic couphng of quarks and gluons is small at such relatwely large distances as the Compton wavelength of the nucleon, the serious question arises, how to confine quarks and gluons To answer th~s question the behawor of the Green functions at large d~stances (infrared region, t -~ ---~) must be studied. According to (4) the effective couphng constant increases as t becomes negatwe. It wdl reach a critical value fCcrtt of the order of one, where perturbation theory breaks down, at the t-value 1 -tent - 2BK

1 2BtCcrrt,

(37)

wtuch corresponds in coordinate space to the critical distance len t = ~,p exp -2]/K 2BKcn t

'

where Xp xs the Compton wavelength o f the nucleon It is conceivable that quarks and gluons get confined at a distance gwen by lcnt. If this Is true, only possibdlty A (K sizeable) makes sense For example for ~¢~- ¼ and ren t ~ 1 one finds lent ~ 8 Xp for ordinary quarks which is about the distance gwen by the meson cloud surrounding the nucleon Provaded this mterpretaUon of the confinement ~s right, logarithmic corrections to scahng must soon show up m the expertments Note however, that the interpretation above Is already excluded ff (32) Is taken seriously (K < ~; lmphes lent > 500 ~.p ))

H Fntzsch, P Mmkowskt, Strong mteracttons

373

It is quite possible that the confinement proceeds through a mechanism, which is nonanalytic in the effective coupling constant and is not seen in any finite order of perturbation theory*. It can arise, perhaps, by the exchange of many (infinitely many) soft gluons. Then the effective couphng constant could be very small even at large distances, and we can accept possibility B In tlus case the critical length (24) cannot be interpreted anymore as the distance at which quarks and gluons are confined Its physical meaning becomes rather unclear, it may play the role of an infrared cutoff As r ~ 0, lcrtt soon gets very large on a macroscopic scale Perhaps there is some reason to identify l ~ t vath the radius of the universe, deterrmned by llubbles constant to R c = 1028 cm. Then one finds for ordinary quarks K- I ~_

Rc

log ~ V = 137.5,

(39)

which is astonishingly close to a -1 *. Takang (39) hterally one has to accept the idea that K "--ct is not a umversal constant, but depends on the present size of the universe and therefore on time logarithmically Independently o f the numerical coincidence (39), one should take seriously the posslbthty that the basic quark-gluon couphng is of the same order as the electromagnetic coupling. In particular, strong, weak and electromagnetic interactions may be described by a single nonabehan gauge field theory* For discussions and encouragement we are grateful to M Gell-Mann. We also would like to thank R. Brandt and lq. Klelnert for discussions After completion of this work we received a prepnnt by H. Pohtzer (Harvard preprlnt), in which ideas related to some of ours were discussed.

References [1] M Gell-Mann and F Low, Phys Rev 95 (1954) 1300 [21 C G Callan, Phys Rev D2 (1970) 1541, K Symanzlk, Comm Math Phys 18(1970)227 [3] D J Gross and F Wflczek, Phys Rev Letters 30 (1973) 1343, H D Pohtzer, Phys Rev Letters 30 (1973) 1346

* This sRuatlon rermnds us of a slmdar one m the theory of superconduchvlty There the energy of a bound electron pair Is non-analytic In the electron-photon couphng constant A bound state Is formed always, even if the attractive force Is very small See also Mmkowska, ref [ 12] * Note that the numerical coincidence (39) depends strongly on the fact that the gauge group Is SU(3) and one has nine colored quarks, which gives B = 9/4n ~' Perhaps by a model of the type considered by Pat1 and Salam [ 131 In such a scheme the ratio r/a depends on the underlying unifying guage group and also on the symmetry breaking

374

H Frttzsch, P Mmkowskt, Strong tnteracttons

[4] S L Adler, Phys Rev 177 (1969) 2426, K G Wdson, Phys Rev 1 7 9 ( 1 9 6 9 ) 1 4 9 9 , R J Crewther, Phys Rev Letters 28 (1972) 1421, W A Bardeen, H Fntzsch and M Gell-Mann, Scale and conformal symmetry in hadron Physics (Wdey, 1973), CERN preprmt TH1538 (1972) [5] D J Gross and F Wdczek, NAL-PUB-73/49 TIIY [61 D J Gross and F Wdczek, Pnnceton Unwerslty preprmt (1973) [7] S Adler and W K Tung, Phys Rev Letters 22 (1969) 978, R J a c k l w a n d G Preparata, Phys Rev 185 (1969)1929 [8] D H Perkins, Proc of the 16th Int Conf on high energy physics, Clucago, 1972 [9] J S Poucher et al, SLAC-PUB-1309 (1973), E D Bloom, SLAC-PUB-1319 (1973), Paper presented at the lnt Symposium on electron and photon interactions at high energies, Bonn, 1973 [10] D J Gross and C H Llewellyn Smith, Nucl Phys B14 (1969) 337 [11] A Zee, Phys Rev D7(1973) 3630 [12] P Mmkowski, Nucl Phys B57(1973) 557 [131 J Patl and A Salam, Phys Rev D8 (1973) 1240