Volume 46B, number 1 PHYSICS LETTERS 3 September 1973

B R O K E N C O N F O R M A L SYMMETRY AND H A D R O N I C SCALING*

G. DOMOKOS, S. KOVESI-DOMOKOS and B.C. YUNN Department o f Physics, Johns Hopkins University, Baltimore, Maryland 21218, USA

Received 16 March 1973

High energy, large momentum transfer hadronic reactions are studied in the framework of broken conformal symmetry. Hadrons lie on almost linear Regge trajectories. Inclusive cross section ~cale as ~ s -2 (q~/s) -4. Large rates.of heavy particle production are predicted.

Summary. Cross sections of inclusive reactions at high energies and momentum transfer appear to have simple "scaling" properties. While this has been known for some time and extensively discussed for lepton- induced processes, scaling in purely hadronic reactions (e.g. in p + p ~ 7r + X) has been observed only recently [1]. Various "parton" models [e.g. 2] have been quite successful in explaining this "hadronic scaling". However, it is quite conceivable that scaling phenomena reflect more general aspects of the dynamics of hadrons that it is suggested by the specific models.

We propose that the hypothesis of asymptotic conformal invariance [3] (together with some addi- tional, rather general, assumptions) leads to an under- standing of at least the main qualitative features ot the data on hadron induced inclusive reactions in the scaling region. The purpose of this note is to sum- marize the first results of such an investigation. A detailed account of the calculations will be published soon.

The general approach to the problem has been described in a previous work [4]. We describe "composite" hadrons by means of effective c-number fields. The restrictions imposed by conformal in- variance are expressed in the form of integral represen- tations for amplitudes (derivatives of the effective action with respect to the c-number fields).

In order to get information about properties of on-shell limits of such amplitudes, we study the breaking of conformal invariance. Treating the aymmetry breaking as a perturbation to the first non- vansihing approximation in the inverse two point function we find hadron states lying on families of approximately linear Regge trajectories. Their inter-

* Research supported by the U.S. Atomic Energy Commission under contract No. AT(11-1) 3285.

cepts are related to the scale dimension. To the same approximation, there occurs no dimension mixing among the states. The symmetry breaking mechanism is tested experimentally by its prediction of total width/mass ratios of resonances. The total theoretical prediction is total width~mass ~ constant, apparently in reasonably good agreement with the data. (See fig. 1) The inclusive cross section for the reaction a + b -+ c + X is estimated in the scaling region. Neg- lecting the spins of the external particle, we find the following asymptotic scaling law:

d3a Eq ~ ' ~ s~(q2/s) -4 FCvx,Y2) , (1)

where a = 1 + ~ ~iPi; 1) i being the scale dimensions of the external fields. (a ~ - 2 if the external fields have nearly canonical dimensions. The function F depends on two dimensionless scale variables, to be defined later. Assuming that F is a slowly varying function,

r~ (6eV ~

£2

l o

. 8

.b

.4

.7.

i~ N

N ~ A

m s(GeV)

Fig. 1. Total widths versus masses squared of N and A reso- nances. Source: Sisding et al., Review of Particle Properties (1972) LBL-100. Resonances marked by appropriate letters, straight line is fitted to the data.

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Volume 46B, number 1 PHYSICS LETTERS 3 September 1973

.10 -

8 :

4"

/ /

,~/ l?) /

/ i "

/ I "

/ . i "

/ . I "

~ (~V9

Fig. 2. Estimated multiplicities of "heavy" particles at large momen tum transfers relative to pions. Dash-dotted curve is average of K, r~, N rates.

one essentially recovers the scaling law of Cline, Halzen and Waldrop [2] (CHW) which gives a reasonably good fit to the ISR data).

Eq. (1), when combined with the approximate expression of the Regge trajectories, leads to the -~- somewhat:bizarre - prediction that the rate of producr;o~ of "heavy'Lparticles relative to pions at large momentum transfers in an increasing function of the energy. (See fig. 2) We find the following ap- proximate formula for the production rates of a particle of mass M relative to pions of mass/a:

R = nM/n ~ ~ K. s (M2-u2)/2 , (2)

where K is some numerical coefficient. (See below.) (A crude estimate leads to the prediction that at the ISR one should see about three times as many "heavy" particles (K, r/, N... ) as pions.) This last pre- diction appears to be independent of the particular process considered, therefore it may be interesting to test it e.g. in lepton induced reactions as well and at several (sufficiently high) energies.

Regge trajectories from broken con formal sym- metry. Consider a theory which is asymptotically invariant under conformal transformations. Whatever is the structure of the "fundamental" dynamics, one can introduce effective fields which describe the towers of resonances in the conformal limit. The one-particle- irreducible amplitudes are given as functional derivatives (with respect to the effective fields) of a conformal invariant effective action [5]. These functional

G-~= , * 4" A 4- • - •

Fig. 3. "Tadpole" expansion of the inverse Green 's function.

derivatives can be constructed by the application of a set of diagram rules, as explained in DK. Consider a simple model, consisting of two interacting effective fields (in what follows, "field", for simplicity), ~ X , x ) and X(X), where X = xUo~ is a coordinate (written in spinor form) and x is an Gel fand-Naimark spin variable. It is implied that tp is a Boson field with 211 = 2]' 2 = o, conforrnal weight r, whereas X is a scalar field of weight d. Let the effective action be W(¢, ×). The inverse Green function, of the field ~p is given by:

G- l ( xx ,X ' x , lX)-- ) . (3)

Eq. (3) can be expanded in a functional Taylor series in powers of the field X, viz:

G - 1 = _ 8 2 W

__ f d4yx(y ) ~3W &p(X,x)8¢(X' ,x ' )Sx(Y) " (4)

The functional derivatives in (4) are taken at × = 0. According to DK, the coefficients of the functional Taylor series (4) can be visualized in terms of confor- mal diagrams. (Essentially the same formalism can be used to describe a Fermion tower of IJl -J2L = ~ in- teracting with the scalar field, X- In the simplest pos- sible version of the theory, the expressions correcpond- ing to the diagrams in fig. 3 have to be multiplied by

1 (7 u ajax u - 7u ~laxu), where the 7u are Dirac matrices. This corresponds to a relativistic "orbital excitation model", cf. Mansouri [6] ).

Suppose now that conformal invarianceois "spon- taneously broken", i.e. the equation (5 W/SX)~.o=O = 0 has a non-trivial, stable solution for Xo, the vacuum expectation value of X. By studying simple models (like the Freund-Nambu model [7]), one conjectures that X has a scale dimension (or conformal weight) d ~ - 2 . In what follows, we write d = - 2 + 2 5 . At

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V o l u m e 46B, n u m b e r 1 PHYSICS L E T T E R S 3 S e p t e m b e r 1973

present we have no reliable method to calculate 8, the anomaly of the dimension of X. Various model calculations suggest that 8 is nonvanishing but small. By redefining × in the usual way (say, × = ×' + ×o) one finds from eq. (4) that the symmetry breaking contri- bution to the inverse Green's function of ~0 is given by the series of conformal diagrams shown in fig. 3). One has reason to believe that a perturbation series in powers of ×o makes sense for G - l , whereas it almost certainly fails for G. In fact, in any physically reasonable theory, the Fourier transform of G possesses singular surfaces corresponding to physical states of the system. Any perturbation expansion necessarily breaks down at such surfaces. The latter, however, correspond to regular null surfaces of G - I , therefore a perturbative estimate of G -1 may give meaningful information about the spectrum. Con- sider in particular the zeroth and first order terms of the "tadpole expansion", fig. 3. A straightforward application of the diagram technique of DK leads to the following expressions:

G~l(p, x, 7"; p', x', r') =

- (2rr)gs(p-p') [8 (r - r')F(o, r, x, x', p)

, 2) ] +xoC(r , r ' ,d )F(o , t , x ,x ,p) + O(X , (4)

where

C(r, r ' , d) =

2 d+r+r' P(d+ 2) 1"(t ( r ' - r - d)) I ' ( t ( r - r ' - d))

X [P(-d)P(~(d-r-r '))P(~(d+r'+r+4);] -1 ,

F(o,'r,x,x',p) = ( 2 1 0 - 4

xf d4x e x p ( - ipx) [(n • x) (~ .x) ] ° ( - x 2 - i0) r .

Here t = ~ ( r+ r '+d ) , and nu = ~ou~ '+, with ~ = (x, 1), ~' = (x' , 1). It is evident from (4) that dimension mixing occurs already at O(Xo), whereas within the symmetry breaking scheme studied, there is no mixing in o. Within this scheme at least, 6 = 0 is not allowed, since then C = oo. Eq. (4) is expanded according to the representations of the little group o f p u by taking

+1

- f d cos6 P,(o, T,p) ¢,(cos6) x ' ,p) -1

in the rest frame o f p .. The angle 0 is defined by cos0 = (k . k ') (Ikl" Ik¢]) -1 in this frame, where

+ r t r+ . . k~ = ~ou~ , ku = ~ ou~ . The integral (5) IS a standard one:~ ~ i f / a n d v = r - o are positive integers. Repeated application of Carlson's and Hartog's theorems per- mits one a continuation in l and v to a sufficiently large domain. According to standard perturbative inversion formulae, the null surfaces of Gol 1 are given by

el(o, + , = 0+O(x2o). r,p)+ ×oC(r,r,d)Fl( o,r dl 2,p)

Near p2 =_ s = 0, the equation of the "leading" null surfaces is of the form:

l+ 1 + v+ n +(-s) 1-a = O, (6)

with n = 0, 1,2,...; some numerical coefficients have been absorbed into the scale of s, originally deter- mined by Xo. To O(Xo), the eigenfunetions are just the unperturbed ones. For fixed v, eq. (6) gives rise to a family of Regge trajectories. I f 8 is small, they are almost linear. The total widths of resonances along the trajectory (6) are given by: 31/2 r = rras + O(82). A fit to nucleon and A-resonances (fig. 1) gives 8 ~ 0.04. Hence our scheme is at least qualitatively consistent with the data. For fixed l, one gets relations between v (the scale dimension of the field) and the masses of the physical states. ("Dimensional trajectories" [8]). In particular, the breaking of SU(3) should reflect itself on conformal quantum numbers. One gets, for instance, from (6)

2 etc. the approximate relations: v K - vw ~ m2K - m~r (This form of the relation is valid if 8 '~ 1 .)

Inclusive cross sections. We estimate these under two simplifying assumptions, a) We neglect all spins (so we work with a Symanzik representation [9] in- stead of the one developed in DK.) b) We work to zeroth order in Xo, consistently with the first order calculation of the spectrum. The inclusive cross section is given at high energies by Ed3o/d3p (2s) -1 A+, where A+ is the absorptive part of the

-);l F(o,r,x,x',p) is expressible through a Jacobi function.

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Volume 46B, number 1 PHYSICS LETTERS 3 September 1973

I

X

Fig. 4. Conformal graph of the six-point function contributing to a+ b ~ c+ X. Momenta are denoted as used in the text.

conformal diagrankshown in fig. 4. We impose the covariance conditions for the spinless case, vi+4 + E v/k = 0 (i, k = 1,2,...,6) written down in ref. [9] and DK by the following trick. Without satisfying the covariance equations, we may write the integral representation for the amplitude F in the form:

F = f . ~ d4xjexp(i ~ kj'x]) f [ - I drik G(rik ) ]

x f [ - [ dUikU~ rat-1 e x p ( i ~ 2 x~u~) ,

where x 2 = (xi--Xk)2. (The integrals over the r~ run along the principal series.) Given a function, f(x), its "Mellin transform", Fa(x) = f~ duu-a-l f(ux), is a homogeneous function of degree a. Therefore, we multiply each u~ in (7) by ?@k and take the multiple Mellin transform:

d;ki v 4 F=f~ -~[X i- r- /~(k 1 ..... )k6) "

The amplitude so obtained evidently satisfies the covariance conditions. A simple substitution of variables brings F to the form:

I'r-r dPi vff4 F= f I - I d4x exp(i~[~ k 'x) J l l--~i p i

dv ac Xik ~l.til2 k } x fI-I-~- a(%) exp {-iE 2v

(Here G is the inverse Mellin transform of G.) After

factoring out the energy-momentum conserving 5. function, viz. F = 3(Ek)A, A can be written as

A = f d 4 x e x p ( i ~ k x ) f ~ -~ildi dpi vf*4

x G(v#c ) exp {-i ~ x~Qg~Xku } .

Integration over the x i gives:

dPi ui+4

x f [7 do__._~ G(vgc) (DetQ)- 2 exp( i '~ k~Q~ 1 kku ) vac

(The matrix Q~ is a function of the Pi and vat.) The 5 X 5 matrix Q~ is inverted approximately by ex- panding it in powers of V/k around - say - va: = 1. As a first estimate, we neglect higher powers of(v~c- 1). Omitting mass terms and introducing the variables Pi = v- t / 2 ~i (with E ~i = 1), the absortive part of the amplitude is finally brought into the form:

dv~ .... ,2+V6)j l j/ ' l-I vg¢ A+ ~B6(2+v 1

× G(vg¢) F(I+~) × (9(Evgcki'kk), (8)

Here ~ = 2 + ~ E vi, BN(Z 1 ..... ZN) = (NN1 r(Zi)) x (r(Xzi))-x.

A glance at eq. (8) shows that A+ obeys the scaling law:

A+ = fi f(y l,Y2) (9)

with s = (pl+P2) 2 ~ 2PlP2,Yl, 2 = s-l(pl,2--q) 2. (This can be shown to be independent of the approxi- mations made inverting the matrix Q.) Further, by working out the kinematics, one finds that Zv~k i. k k is of the form:

E va~ki'k k ~" s[(023-021-035+o45 )

+Yl(Vls+v26-v12-o56) + Y2(V14 +v36-v13-v46 )] Without loss of generality we may write G = llvac expH(v~), where H(vg~) may be e.g. finite at v#c = 0. Introducing some new integration variables by w12 =Y 1Vl 2, etc., we see that if, for instance, H is a

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Volume 46B, number 1 PHYSICS LETTERS 3 September 1973

slowly varying function:~: around V/k = 0, we essenti- ally recover the CHW scaling law, viz. E d 3 o / d 3 q

s - l + # ( y l ' Y 2 ) - 4 . The exponent/3 is in fact close to ( - 1);/3 = - 1 would be obtained if all the v i were canonical (v i = - 1). Connection with CHW is established by noticing that in terms of CMS variables Y l Y 2 ~ q 2 s - 1. From the scaling law (9), after inte- grating over momenta , one obtains the relative multi- plicities of various masses in the form:

R = nM/n ~ ~ K" s (vM-vl~)/2 . (1 O)

Eq. (2) is obtained from this by taking into account the relation between dimensions and masses of particles.

The curves in fig. 2 were calculated under the as- sumption that the integral in (8) is insensitive to small variations of/3 around/3 ~ - 1. (This is approxi- mately true, since near/3 = - 1, O(~c)xt~l-'( 1 +/3)-1 ~, 6(x).) Thus K is given by the ratio of two polybeta functions, B 6. (Product ion rates for nulceons are probably lower when the spin of the nucleon is tzken into account; the curve shown in fig. 2 applies - strictly speaking - to a fictitious, spinless nucleon.)

Comments . Broken conformal invariance appears to have a considerable predictive power concerning high energy, large momentum transfer hadronic processes. I t is evidently clear to the reader that one is still far from a consistent theory. Yet, these - ad- mit tedly naive - estimates lead one to conjecture that fine details of the dynamics of complicated systems (as hadrons appear to be) may be relatively unimportant as far as practically observable propert ies are concerned. From the experimental point of view,

:~2 This approximation is not valid near the boundary of phase space.

the reasoning which leads one to eq. (10) suggests that at sufficiently high energies and large momentum transfers, one should see an increasing number of low- spin, heavy particles produced in any reaction. While this predict ion is somewhat startling in view of the overwhelming dominance of pions at low momentum transfers, it does not seem to contradict any physical principle or experimental result known to these authors. Should it be borne out at least qualitatively by future experiments, it would strengthen one's confidence in the scheme sketched above.

References

[1] CCR collaboration, contribution ot the NAL-Chicago Conference (September, 1972).

[2] S.M. Berman, J.D. Bjorken and J.B. Kogut, Phys. Rev. D4 (1971) 3388; D. Horn and M. Moshe, Nucl. Phys. 48B (1972) 557; J.F. Gunion, S.J. Brodsky and R. Blankenbecler, Phys. Rev. D6 (1972) 2652; D. Cline, F. Halzen and Waldrop, Wisconsin preprint (1973)i P.V. Landshoff and J.C. Polkinghorne, Cambridge preprints (1972), DAMTP 72/43, 72/48.

[3] For recent reviews cf. S Ferrara, R. Gatto and A.F. Grillo, Frascati report LNF- 71/79 (1971). Springer Tracts in Modern Physics, Vol. 67, Springer Verlag, Berlin; G. Mack, Kaiserslautern Lectures (1972).

[4] G. Domokos and S. K6vesi-Domokos, JHU preprint (1972) COO-3285-2. (Hereafter quoted as DK.).

[5] G. Jona-Lasinio, Nuoco Cim. 34 (1964) 1790; B. Zumino, in Lectures on Elementary particles and quantum field theory, 'Vol. 2, eds. S Deser, M. Grimm and H. Pendleton (MIT press, 1970).

[6] F. Mansouri, Nuovo Cim. 3A (1971) 220. [7] P.G.O. Freund and Y. Nambu, Phys. Rev. 174 (1968)

1741;Cf. also B. Zumino, ref. [5]. [8] E. Del Giudice et al., Nuovo Cim. 12A (1972) 813. [9] K. Symanzik, Lett. Nuovo Cim. 3 (1972) 734.

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