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Testing gauge theories of weak interactions in the deep inelastic region
Volume 45B, number 3
PHYSICS LETTERS
23 July 1973
TESTING GAUGE THEORIES OF WEAK INTERACTIONS IN THE DEEP INELASTIC REGION S.P. DE ALWlS* Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, UK Received 22 September 1972 Revised manuscript received 26 October 1972 The analogs of the Adler sum rule and the Llewellyn-Smith relation for the Georgi-Glashow and LeePrentki-Zumino models for weak interactions, are derived. It is suggested that these could provide useful tests of the models. There are by now several [ 1 - 3 ] unified renormalizable models of weak and electromagnetic interactions based on the Higgs-Kibble idea of a spontaneously broken gauge theory. The earlier models [1] are based on the gauge group SU(2) ® U(1) and thus have four generators, two of which are the usual charges, the other two being combinations of the electric charge and a hitherto unobserved, neutral weak charge. This neutral weak charge has a neutrino term and the upper bounds on various neutrino induced reactions such as vu + e -* vu + e and v~ + p -~ vu + p + n ° may well rule these models out [4]. Thus it is of interest to study models in which a) the neutral current is absent altogether - i.e. that of Georgi and Glashow [2] (G.G.) or b) the neutral current appears only as a short range parity violating correction to electromagnetism (involves no neutrino term) - i.e. that of Lee Prentki and Zumino [3] (LPZ). However both models accomplish these feats at the price of introducing heavy leptons. The GG model has two electronic (X +, X °) and two muonic ( y + y o ) heavy leptons. The neutral lepton may be difficult to detect but the Y± pair may be seen in colliding beam or 3' nucleus scattering experiments. Furthermore the bounds on the anomalous magnetic moment of the muon gives m y , < 5 GeV [5, 6], so that it may be possible to, detect it, at least in the scattering experiments - in the near future. However even if it is detected, although it would be evidence in favour of models without neutral currents, in the absence of lower bounds on the heavy leptons in other models of this sort such as the LPZ model (and its extensions to larger gauge groups)" it would not be a test capable of distinguishing between them. Calculating the higher order weak corrections to various processes will naturally give results which are of the expected order of magnitude, and although this is aesthetically satisfying, it is no real rest of the theory, both because present experiment fits the lowest order calculations pretty well and even if experimental accuracy is improved one still has to (at least in some cases) include the hadronic effects (to the same order) before one can compare the results. There are however certain other useful tests which may be made following from the fact that the form of the weak interaction theory imposes constraints on hadronic structure. For instance, the GG model predicts eight quarkst I and thus there would be 128 vector and axial currents satisfying a U(8) × U(8) algebra at equal times and furthermore assuming canonical light cone structure, an asymptotic (on the light cone) U(8) × U(8) symmetry [for LPZ U(6) × U(6)]. These constraints lead to sum rules corresponding to the Adler [7] sum rule and the Llewellyn-Smith [7] relation in the usual quark model. The purpose of this note is to derive these for the GG and the LPZ models. Now, of course, both models (GG and LPZ) predict the existence of charmed hadronic states, and the energies at which the sum rules should be tested must lie above the threshold for charmed particle production, Theory Division, CERN 1211 Geneva 23, Switzerland. 1 The original five-quark version has been ruled out since the predicted rate for KL~ t~ is far too high [of O(G. c0], [6].
* Address:
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so that in principle one could test the models by directly searching for the.charmed states. However, even though they decay only through the weak interactions, because of their high mass they would have a very short lifetime (~ 10-13 sec for masses ~ 5 GeV) and furthermore could decay into large number of channels, thus making their detection difficult. Again, detection of a high mass 'stable' (charmed) state would not by itself be a crucial test of the models considered, nor would it provide an easy means of distinguishing between the different model. In any case, even in the event that the tests proposed here (which involve just the neutrino- and electro-production inclusive experiments) are more difficult to perform than a direct search for charmed states, it would still be of interest to verify them. With an effective Lagrangian for the semi-leptonic weak interactions of the form
(1) where G(~ 10-5M~p2) is th~ Fermi constant and Mw the mass of the charged vector boson, we may define [7] the structure functions for inclusive neutrino scattering on a hadronic target H, v(k)/ff(k) + H(p) -* £(k')[£(k'.) + X by +pup v
- - fflI-u(O)lX}CKI-v. O)lU)(2--384.p+k-P '-k') =-guvW1 ~
a fl
W2-ie #voa _P-q" )AkI2 W3+...
x where Wi =- Wi(q2, v) with v = p- q and q = k - k'. The inclusive differential cross-section is then given to lowest order in G by,
d2ov;~ G2 ME dcody ( 1 - 2 y M E / ~ M w ) 2 7rw2
_y_
I( 1
My
(7~,V+y2 ~__~_1T-y(1 - ~ y ) ~ ~
"
where y = v/ME, ~o = 2 v / - q 2 , Gl(W ' q2[M2 ) = Wl(V ' q2) and G i = vWi[M 2 (i =/=1), E being the incident lab energy of the neutrino. In the Georgi Glashow (GG) model [2] the interaction Lagrangian is given by Lint = e [ l # J -
+
I ¢ ' $ ~ + AuJ~"m"]
(2)
where
d"+= {~HTu(1-')'5)C~(~* H + ½~HTv( 1+")'5)CH(0)*H =-~-sinfl[~r~,~,(l-T5) n cos0 +ff3,u(1-75)X sine] + .... je.m. = ~a3, a ~ a
(3) (4)
#
where ~H = column {p'p n q0 X q0' q - q - ' ) and
]. uT() o
c(t3)=
o _
u'T(~)
(5)
O
where in the limit 0 = 0 (we will be interested only in the strangeness-conversing processes)
p, n, ?~are an SU(3) triplet with charges (1,0, 0); p, n being the isodoublet, p', q0, q0', q - and q - ' are unitary singlet quarks whose presence is dictated by the various requirements of the model [2, 5, 6]. The strong interac254
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tion terms are taken to be chiral U(8) invariant with the observed hierarchy of symmetry breaking given by the mass terms? 2. In the second equality of eq. (3) the dots indicate the part of the weak current which causes transitions to 'charmed' hadron states and the term in square brackets is the usual (low energy) weak interaction current. Thus comparing with (1) we have GM2w/Vr2= ¼e2 sin2/3 and J ~ = 2 sin- 1/3J~- . Thus I4u~v= 4 sin-2t~f ~ - ~ e x p ( i q • x) ffI(p)l [ J ~ ( x ) J - ( 0 ) ] IH(p))
(6)
d4x = fa --~-~-"exp(iq • x) [(H(p)I [J~AC=0(X), j-AC=0(0)] IH(p)) + (I-I(p)l [J~AC*0(X), Jv6C~0(0)] ill(p))] = - g u y Wl + where j~C=O is the usual weak current and j,~C~O is the current which changes charm and creates the served charmed hadron states. Now the model has the equal time current algebra j+
-
[ 0(x),J0(0)]Xo=0 =J~'m'(o)63(x)
as
yet unob-
('.'[C~,CH] = Q)
and thus the Adler sum rule [7] turns out to be :
[G~(co, q 2 ) _ G~(t.o, q2)] den = 4 sin -2/3 (Q) 69
(7)
1
where (Q) is the charge o f the target'f 3 . This is to be contrasted with 4(/3) for the right-hand side in the usual Cabibbo theory. This sum rule may be used to estimate the free parameter 13. [The only other method o f determining/3 is to estimate the mass of the W-boson. Indeed the original (incorrect) estimate of an upper bound for M w in the five-quark model implied a/~ < 0.08 leading to the unbelievable result that for protons the right-hand side o f ( 7 ) was > 6 2 8 ! ] Furhter tests of the model may be done in the deep inelastic region provided one assumes a) that the strong interactions are 'softened' in the sense that the leading light cone singularities of current commutators are canonicalS. [9], implying that the structure functions Gf(co, q2) scale, (for large q2 and fixed co), and, b) that these leading singularities have the same structure as would be indicated by calculating them in a non-interacting model [9] (or a gluon model treated formally'P). The observed sealing for the structure functions of electroproduction a-nay be taken as evidence for the former assumption, but for the latter the evidence is so far almost non-existentt s . It is however an elegant and well-accepted hypothesis and we will make it in order to derive a further test of the model. If we consider U(8) currents a~ (r, x ) = ~HTta(1 - r3,s)~Ai~0H where the A i are a set of U(8) matrices normalized such that I'r(AiA/) = 260., and r = -+1, then we may define F ~ , 3 ( 6 o ) by the equations [9]
f._~_~_ d4x exp(u?x) ,. . (pl [.~(r, x)Jv/(r,
0)] Ip) =
PuPv -guvG~(6o,"q2) + G~(~, u
q2) _
i r eu~opaqO 2v
G~(co, q2)
t 2 The assumption that only the mass terms break the U(8) × U(8) symmetry is the most natural one but it is not a requirement of the gauge symmetry of the weak interactions. Of course for deriving light-cone sum rules we require later on [assumption b) below] asymptotic U(8) X U(8) invariance, and the above assumption on the form of the symmetry breaking is the simplest one which would lead to this• Needless to say, the Adler sum rule for the model is independent of this assumption. t 3 It needs hardly be stressed that (6) involves summing over all (i.e. charmed and uncharmed) states, as would be the ease in an inclusive experiment. If one restricted the final states to uncharmed ones the usual Adler sum rule would of course be obtained. Moreover it should be noted that the charm changing parts of the weak current are not charge symmetric so that the usual • i,' P relations G_V ---Gn, Gp = GVn are not valid. ?4 Alternatively one may use a patton model approach [ 7]. ?s Of course any verification of this assumption would also be a test of the model used in calculating the various relations that are to be tested• 255
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and F~i,2,3(co) = limq2_,_. G~,2,3(co, q2). Our assumption a) implies the existence of the above limit and from assumption b) we have further that [9], where
(8a,b,c)
= Pu ± f e x p ( - i ~°x 1 "p) (pl~(½x) 7u(1 - r'Ys)½Ak ~k(-½x) +,(x ~ - x ) lp)
(9)
and =¼rr [Ai, A i l h k ,
dt~=¼ rr (Ai, A/)A k .
(10)
We also note that on the light cone
[Jiu(r=+l,x),Jiv(r =-I,0)]
=0.
(11)
Now using these equations one may derive all the sum rules that have been derived for the ordinary quark model, but not all of these are of much use in testing the weak interaction theory that we are interested in. Thus in the GG model the gauge symmetry renormalizability and the absence of strangeness-changing neutral effects require one to have eight quarks with charge Q = diag [ 1, 1,0, 0, 0, 0, -1, -1 ] and given that the strong interactions conserve one SU(2) (isospin) symmetry we require just one isodoublet (pn quarks) the rest being isospin singlets. Thus the isospin matrix is taken to be ½A3, where A 3 = diag [0, 1, -1,0, 0, 0, 0, 0]. The Iwpercharge assignment is, however, non-unique, and furthermore the model has five more conserved (by the strong interactions) quantum numbers (five 'charms') and these assignments are also non-unique, so that clearly only that relation which depends on the isospin biloeal operator, i.e. the Llewellyn-Smith (L-S) relation ]7] for the model, is a test of the GG theory in so far as it is independent of the hypercharges and charms of the quarks. Let us evaluate this sum rule. From (5) we have CH ] = Q ,
(C~,CH)=diag[1,2V, 2V, 1] ,
rr CHC~_I= 4,
where { sin23 sin3 cos3 cos2 ] V = \sin3 cos3 so that
ifc~{CH3 = ¼ , ifQQ3=0, dc~iCH3 = ¼(1-- 2 sin2(3) , dQQ3=½. Thus from (6) (and the corresponding one for electro-production) and eqs. (8)-(11) we have coF~h =B_V3 +(I = 0 terms),
coF~h = 2 sin-23B+V3 + 2(sin-23 - 1)B_V3 +(]= 0 terms),
F~H = - 2B+V3 + ( I = 0 terms), where 2 8 v 3 _- B ±3( + , . co) + B ±3( - , . c o ) . Hence we have the relation F~ h - F ~ = 2co(1- sin23)(F~h - F : ) -
co sin23(F~h-F2~)
(12)
h and h being any hadronic isodoublett 6 • t 6 Perhaps it is worth noting that the relations (7) and (12) are the same as for the five-quark case. For the Adler sum rule this is obvious, and for the LleweUyn-Smithrelation it is partly a consequence of the fact that the extra quarks are isosinglets.
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Let us now examine the model of Lee, Prentki and Zumino [3] (LPZ). In this model there are six quarks? 7 [the usual SU(3) triplet p, n, ), and three additional unitary singlets p', q - and q-'. Again we arrange these in a unitarity (this time U(6)) sextet representation t
~H =
P P X n __s
The interaction Lagrangian is
Lint = g { W J ~ + I4I+ Jt`+ +A~J;- } +g'A~.~ with Ju+=~r~HVt`(I-75)C~
a ,
-/~± =½~H7t`( 1 + 7 5 ) Q ~ H •
The electromagnetic current is J~ezn.= ~H7t`Q~H with the charge e = gg,/(g2+g,2)t/2, the photon being At` = A t, sin0w +A ~ cos0 w, 0 w = arctang'/g. The matrix CH is defined by CH =
0
0
-U
0
where
u=
L sinO
cosO J
and the charge matrix Q = diag [1 1 0 0 - 1 - 1 ] . ExpandmgJ~ we have •
÷
JU÷ = ½PTt`(1-75) n cos0 + ½PTt`(1-75) n sin8 + ... (the dots representing terms which cause transitions to charmed states) and comparing with (1), G M ~ v / V ~ = g/2, j w = 2J~. The model has the equal time current algebra, [J;(x), Jo(O)] = ½J~e.m. - ½-~757t`Q~ , so that the Adler sum rule [7] turns out to be
?[G; (co, q 2 ) _ a ~ c o , q 2 ) l
dco -~--_ 2
(13)
1 For scattering on protons this is indistiguishable from the conventional (V - A) theory but is can be easily tested on nucleii. ~he light cone algebra can be treated as before and corresponding to the unique assignment of charge and isospin one has the sum rule (F3UH - F~'H) = --co(F~H - F~H).
(14)
Now of course, as we pointed out in our introductory remarks, the sum rules (12) and (14) are valid only in the scaling region which would set in above the threshold for charmed particle production. Thus if (12) or (14) is verified in the presently observed scaling region, say for m i s ~ g masses greater than about 2.5 GeV, this would mean that the charmed states are being produced and that their masses are less than 2.5 GeV. On the other hand
?7 Actually in order to construct the baryons, a seventh quark must be added, but since it does not enter in the weak or electrompgnetic current it is irrelevant for out purposes.
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it is conceivable that the charmed states are at a much higher mass and the present scaling region is an intermediate one in which the relations (12) and (14) would be replaced b y one corresponding to an " a s y m p t o t i c " U(3) × U(3) symmetry generated b y the three p, n,. ~ SU(3) triplet quarks, i.e. the relation one obtains in the Sakata m o d e l t s
When the charmed particle threshold is reached however there should be a breaking of scaling and then a new scaling regime should set in which the relations (12) or (14) would be valid. I wish to thank Professor K. Bardakci for a critical reading o f the manuscript. t s Of course the other relations which follows from the Sa~katamodel would not be obtained in this case.
References [1] S. Weinberg, Phys. Rev. Lett. 19 (3967) 1264; 27 (1971) 1688; A. Saiam, in Elementary particle physics, ed. N. Svartholm, (Almqvist and Wiksells, Stockholm, 1968) p. 367. [2] H. Georgi and S.L. Giashow, Phys. Rev. Lett. 28 (1972) 1494. [3] B.W. Lee, Phys. Rev. D. to be published. [4] J. Prentki and B. Zumino, Nucl. Phys. B47 (1972) 99. [5] K. Fujikawa, B.W. Lee and A.I. Sanda, preprint NAL-THY-55 (1972); J.R. Primack, Talk given at the 16th Int. Conf. on High-Energy Physics, Batavia, I11.,Sept. 1972. [6] B.W. Lee, J.R. Primack and S.B. Treiman, preprint NAL-THY-74 (1972). [7] C.H. Llewellyn-Smith, Neutrino reactions at accelerator energies, Physes Reports 3 (1972) 261. [8] S.L. Giashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [9] H. Fritzsch and M. Gell-Mann, Talk presented at the Int. Conf. on Duality and Symmetry, Tel-Aviv, Israel (Weizmann Scienee Press, 1971 ); D. Gross and S.B. Trelman, Phys. Rev. D4 (1971) 1059.
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TESTING GAUGE THEORIES OF WEAK INTERACTIONS IN THE DEEP INELASTIC REGION S.P. DE ALWlS* Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, UK Received 22 September 1972 Revised manuscript received 26 October 1972 The analogs of the Adler sum rule and the Llewellyn-Smith relation for the Georgi-Glashow and LeePrentki-Zumino models for weak interactions, are derived. It is suggested that these could provide useful tests of the models. There are by now several [ 1 - 3 ] unified renormalizable models of weak and electromagnetic interactions based on the Higgs-Kibble idea of a spontaneously broken gauge theory. The earlier models [1] are based on the gauge group SU(2) ® U(1) and thus have four generators, two of which are the usual charges, the other two being combinations of the electric charge and a hitherto unobserved, neutral weak charge. This neutral weak charge has a neutrino term and the upper bounds on various neutrino induced reactions such as vu + e -* vu + e and v~ + p -~ vu + p + n ° may well rule these models out [4]. Thus it is of interest to study models in which a) the neutral current is absent altogether - i.e. that of Georgi and Glashow [2] (G.G.) or b) the neutral current appears only as a short range parity violating correction to electromagnetism (involves no neutrino term) - i.e. that of Lee Prentki and Zumino [3] (LPZ). However both models accomplish these feats at the price of introducing heavy leptons. The GG model has two electronic (X +, X °) and two muonic ( y + y o ) heavy leptons. The neutral lepton may be difficult to detect but the Y± pair may be seen in colliding beam or 3' nucleus scattering experiments. Furthermore the bounds on the anomalous magnetic moment of the muon gives m y , < 5 GeV [5, 6], so that it may be possible to, detect it, at least in the scattering experiments - in the near future. However even if it is detected, although it would be evidence in favour of models without neutral currents, in the absence of lower bounds on the heavy leptons in other models of this sort such as the LPZ model (and its extensions to larger gauge groups)" it would not be a test capable of distinguishing between them. Calculating the higher order weak corrections to various processes will naturally give results which are of the expected order of magnitude, and although this is aesthetically satisfying, it is no real rest of the theory, both because present experiment fits the lowest order calculations pretty well and even if experimental accuracy is improved one still has to (at least in some cases) include the hadronic effects (to the same order) before one can compare the results. There are however certain other useful tests which may be made following from the fact that the form of the weak interaction theory imposes constraints on hadronic structure. For instance, the GG model predicts eight quarkst I and thus there would be 128 vector and axial currents satisfying a U(8) × U(8) algebra at equal times and furthermore assuming canonical light cone structure, an asymptotic (on the light cone) U(8) × U(8) symmetry [for LPZ U(6) × U(6)]. These constraints lead to sum rules corresponding to the Adler [7] sum rule and the Llewellyn-Smith [7] relation in the usual quark model. The purpose of this note is to derive these for the GG and the LPZ models. Now, of course, both models (GG and LPZ) predict the existence of charmed hadronic states, and the energies at which the sum rules should be tested must lie above the threshold for charmed particle production, Theory Division, CERN 1211 Geneva 23, Switzerland. 1 The original five-quark version has been ruled out since the predicted rate for KL~ t~ is far too high [of O(G. c0], [6].
* Address:
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so that in principle one could test the models by directly searching for the.charmed states. However, even though they decay only through the weak interactions, because of their high mass they would have a very short lifetime (~ 10-13 sec for masses ~ 5 GeV) and furthermore could decay into large number of channels, thus making their detection difficult. Again, detection of a high mass 'stable' (charmed) state would not by itself be a crucial test of the models considered, nor would it provide an easy means of distinguishing between the different model. In any case, even in the event that the tests proposed here (which involve just the neutrino- and electro-production inclusive experiments) are more difficult to perform than a direct search for charmed states, it would still be of interest to verify them. With an effective Lagrangian for the semi-leptonic weak interactions of the form
(1) where G(~ 10-5M~p2) is th~ Fermi constant and Mw the mass of the charged vector boson, we may define [7] the structure functions for inclusive neutrino scattering on a hadronic target H, v(k)/ff(k) + H(p) -* £(k')[£(k'.) + X by +pup v
- - fflI-u(O)lX}CKI-v. O)lU)(2--384.p+k-P '-k') =-guvW1 ~
a fl
W2-ie #voa _P-q" )AkI2 W3+...
x where Wi =- Wi(q2, v) with v = p- q and q = k - k'. The inclusive differential cross-section is then given to lowest order in G by,
d2ov;~ G2 ME dcody ( 1 - 2 y M E / ~ M w ) 2 7rw2
_y_
I( 1
My
(7~,V+y2 ~__~_1T-y(1 - ~ y ) ~ ~
"
where y = v/ME, ~o = 2 v / - q 2 , Gl(W ' q2[M2 ) = Wl(V ' q2) and G i = vWi[M 2 (i =/=1), E being the incident lab energy of the neutrino. In the Georgi Glashow (GG) model [2] the interaction Lagrangian is given by Lint = e [ l # J -
+
I ¢ ' $ ~ + AuJ~"m"]
(2)
where
d"+= {~HTu(1-')'5)C~(~* H + ½~HTv( 1+")'5)CH(0)*H =-~-sinfl[~r~,~,(l-T5) n cos0 +ff3,u(1-75)X sine] + .... je.m. = ~a3, a ~ a
(3) (4)
#
where ~H = column {p'p n q0 X q0' q - q - ' ) and
]. uT() o
c(t3)=
o _
u'T(~)
(5)
O
where in the limit 0 = 0 (we will be interested only in the strangeness-conversing processes)
p, n, ?~are an SU(3) triplet with charges (1,0, 0); p, n being the isodoublet, p', q0, q0', q - and q - ' are unitary singlet quarks whose presence is dictated by the various requirements of the model [2, 5, 6]. The strong interac254
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tion terms are taken to be chiral U(8) invariant with the observed hierarchy of symmetry breaking given by the mass terms? 2. In the second equality of eq. (3) the dots indicate the part of the weak current which causes transitions to 'charmed' hadron states and the term in square brackets is the usual (low energy) weak interaction current. Thus comparing with (1) we have GM2w/Vr2= ¼e2 sin2/3 and J ~ = 2 sin- 1/3J~- . Thus I4u~v= 4 sin-2t~f ~ - ~ e x p ( i q • x) ffI(p)l [ J ~ ( x ) J - ( 0 ) ] IH(p))
(6)
d4x = fa --~-~-"exp(iq • x) [(H(p)I [J~AC=0(X), j-AC=0(0)] IH(p)) + (I-I(p)l [J~AC*0(X), Jv6C~0(0)] ill(p))] = - g u y Wl + where j~C=O is the usual weak current and j,~C~O is the current which changes charm and creates the served charmed hadron states. Now the model has the equal time current algebra j+
-
[ 0(x),J0(0)]Xo=0 =J~'m'(o)63(x)
as
yet unob-
('.'[C~,CH] = Q)
and thus the Adler sum rule [7] turns out to be :
[G~(co, q 2 ) _ G~(t.o, q2)] den = 4 sin -2/3 (Q) 69
(7)
1
where (Q) is the charge o f the target'f 3 . This is to be contrasted with 4(/3) for the right-hand side in the usual Cabibbo theory. This sum rule may be used to estimate the free parameter 13. [The only other method o f determining/3 is to estimate the mass of the W-boson. Indeed the original (incorrect) estimate of an upper bound for M w in the five-quark model implied a/~ < 0.08 leading to the unbelievable result that for protons the right-hand side o f ( 7 ) was > 6 2 8 ! ] Furhter tests of the model may be done in the deep inelastic region provided one assumes a) that the strong interactions are 'softened' in the sense that the leading light cone singularities of current commutators are canonicalS. [9], implying that the structure functions Gf(co, q2) scale, (for large q2 and fixed co), and, b) that these leading singularities have the same structure as would be indicated by calculating them in a non-interacting model [9] (or a gluon model treated formally'P). The observed sealing for the structure functions of electroproduction a-nay be taken as evidence for the former assumption, but for the latter the evidence is so far almost non-existentt s . It is however an elegant and well-accepted hypothesis and we will make it in order to derive a further test of the model. If we consider U(8) currents a~ (r, x ) = ~HTta(1 - r3,s)~Ai~0H where the A i are a set of U(8) matrices normalized such that I'r(AiA/) = 260., and r = -+1, then we may define F ~ , 3 ( 6 o ) by the equations [9]
f._~_~_ d4x exp(u?x) ,. . (pl [.~(r, x)Jv/(r,
0)] Ip) =
PuPv -guvG~(6o,"q2) + G~(~, u
q2) _
i r eu~opaqO 2v
G~(co, q2)
t 2 The assumption that only the mass terms break the U(8) × U(8) symmetry is the most natural one but it is not a requirement of the gauge symmetry of the weak interactions. Of course for deriving light-cone sum rules we require later on [assumption b) below] asymptotic U(8) X U(8) invariance, and the above assumption on the form of the symmetry breaking is the simplest one which would lead to this• Needless to say, the Adler sum rule for the model is independent of this assumption. t 3 It needs hardly be stressed that (6) involves summing over all (i.e. charmed and uncharmed) states, as would be the ease in an inclusive experiment. If one restricted the final states to uncharmed ones the usual Adler sum rule would of course be obtained. Moreover it should be noted that the charm changing parts of the weak current are not charge symmetric so that the usual • i,' P relations G_V ---Gn, Gp = GVn are not valid. ?4 Alternatively one may use a patton model approach [ 7]. ?s Of course any verification of this assumption would also be a test of the model used in calculating the various relations that are to be tested• 255
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and F~i,2,3(co) = limq2_,_. G~,2,3(co, q2). Our assumption a) implies the existence of the above limit and from assumption b) we have further that [9], where
(8a,b,c)
= Pu ± f e x p ( - i ~°x 1 "p) (pl~(½x) 7u(1 - r'Ys)½Ak ~k(-½x) +,(x ~ - x ) lp)
(9)
and =¼rr [Ai, A i l h k ,
dt~=¼ rr (Ai, A/)A k .
(10)
We also note that on the light cone
[Jiu(r=+l,x),Jiv(r =-I,0)]
=0.
(11)
Now using these equations one may derive all the sum rules that have been derived for the ordinary quark model, but not all of these are of much use in testing the weak interaction theory that we are interested in. Thus in the GG model the gauge symmetry renormalizability and the absence of strangeness-changing neutral effects require one to have eight quarks with charge Q = diag [ 1, 1,0, 0, 0, 0, -1, -1 ] and given that the strong interactions conserve one SU(2) (isospin) symmetry we require just one isodoublet (pn quarks) the rest being isospin singlets. Thus the isospin matrix is taken to be ½A3, where A 3 = diag [0, 1, -1,0, 0, 0, 0, 0]. The Iwpercharge assignment is, however, non-unique, and furthermore the model has five more conserved (by the strong interactions) quantum numbers (five 'charms') and these assignments are also non-unique, so that clearly only that relation which depends on the isospin biloeal operator, i.e. the Llewellyn-Smith (L-S) relation ]7] for the model, is a test of the GG theory in so far as it is independent of the hypercharges and charms of the quarks. Let us evaluate this sum rule. From (5) we have CH ] = Q ,
(C~,CH)=diag[1,2V, 2V, 1] ,
rr CHC~_I= 4,
where { sin23 sin3 cos3 cos2 ] V = \sin3 cos3 so that
ifc~{CH3 = ¼ , ifQQ3=0, dc~iCH3 = ¼(1-- 2 sin2(3) , dQQ3=½. Thus from (6) (and the corresponding one for electro-production) and eqs. (8)-(11) we have coF~h =B_V3 +(I = 0 terms),
coF~h = 2 sin-23B+V3 + 2(sin-23 - 1)B_V3 +(]= 0 terms),
F~H = - 2B+V3 + ( I = 0 terms), where 2 8 v 3 _- B ±3( + , . co) + B ±3( - , . c o ) . Hence we have the relation F~ h - F ~ = 2co(1- sin23)(F~h - F : ) -
co sin23(F~h-F2~)
(12)
h and h being any hadronic isodoublett 6 • t 6 Perhaps it is worth noting that the relations (7) and (12) are the same as for the five-quark case. For the Adler sum rule this is obvious, and for the LleweUyn-Smithrelation it is partly a consequence of the fact that the extra quarks are isosinglets.
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Let us now examine the model of Lee, Prentki and Zumino [3] (LPZ). In this model there are six quarks? 7 [the usual SU(3) triplet p, n, ), and three additional unitary singlets p', q - and q-'. Again we arrange these in a unitarity (this time U(6)) sextet representation t
~H =
P P X n __s
The interaction Lagrangian is
Lint = g { W J ~ + I4I+ Jt`+ +A~J;- } +g'A~.~ with Ju+=~r~HVt`(I-75)C~
a ,
-/~± =½~H7t`( 1 + 7 5 ) Q ~ H •
The electromagnetic current is J~ezn.= ~H7t`Q~H with the charge e = gg,/(g2+g,2)t/2, the photon being At` = A t, sin0w +A ~ cos0 w, 0 w = arctang'/g. The matrix CH is defined by CH =
0
0
-U
0
where
u=
L sinO
cosO J
and the charge matrix Q = diag [1 1 0 0 - 1 - 1 ] . ExpandmgJ~ we have •
÷
JU÷ = ½PTt`(1-75) n cos0 + ½PTt`(1-75) n sin8 + ... (the dots representing terms which cause transitions to charmed states) and comparing with (1), G M ~ v / V ~ = g/2, j w = 2J~. The model has the equal time current algebra, [J;(x), Jo(O)] = ½J~e.m. - ½-~757t`Q~ , so that the Adler sum rule [7] turns out to be
?[G; (co, q 2 ) _ a ~ c o , q 2 ) l
dco -~--_ 2
(13)
1 For scattering on protons this is indistiguishable from the conventional (V - A) theory but is can be easily tested on nucleii. ~he light cone algebra can be treated as before and corresponding to the unique assignment of charge and isospin one has the sum rule (F3UH - F~'H) = --co(F~H - F~H).
(14)
Now of course, as we pointed out in our introductory remarks, the sum rules (12) and (14) are valid only in the scaling region which would set in above the threshold for charmed particle production. Thus if (12) or (14) is verified in the presently observed scaling region, say for m i s ~ g masses greater than about 2.5 GeV, this would mean that the charmed states are being produced and that their masses are less than 2.5 GeV. On the other hand
?7 Actually in order to construct the baryons, a seventh quark must be added, but since it does not enter in the weak or electrompgnetic current it is irrelevant for out purposes.
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it is conceivable that the charmed states are at a much higher mass and the present scaling region is an intermediate one in which the relations (12) and (14) would be replaced b y one corresponding to an " a s y m p t o t i c " U(3) × U(3) symmetry generated b y the three p, n,. ~ SU(3) triplet quarks, i.e. the relation one obtains in the Sakata m o d e l t s
When the charmed particle threshold is reached however there should be a breaking of scaling and then a new scaling regime should set in which the relations (12) or (14) would be valid. I wish to thank Professor K. Bardakci for a critical reading o f the manuscript. t s Of course the other relations which follows from the Sa~katamodel would not be obtained in this case.
References [1] S. Weinberg, Phys. Rev. Lett. 19 (3967) 1264; 27 (1971) 1688; A. Saiam, in Elementary particle physics, ed. N. Svartholm, (Almqvist and Wiksells, Stockholm, 1968) p. 367. [2] H. Georgi and S.L. Giashow, Phys. Rev. Lett. 28 (1972) 1494. [3] B.W. Lee, Phys. Rev. D. to be published. [4] J. Prentki and B. Zumino, Nucl. Phys. B47 (1972) 99. [5] K. Fujikawa, B.W. Lee and A.I. Sanda, preprint NAL-THY-55 (1972); J.R. Primack, Talk given at the 16th Int. Conf. on High-Energy Physics, Batavia, I11.,Sept. 1972. [6] B.W. Lee, J.R. Primack and S.B. Treiman, preprint NAL-THY-74 (1972). [7] C.H. Llewellyn-Smith, Neutrino reactions at accelerator energies, Physes Reports 3 (1972) 261. [8] S.L. Giashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [9] H. Fritzsch and M. Gell-Mann, Talk presented at the Int. Conf. on Duality and Symmetry, Tel-Aviv, Israel (Weizmann Scienee Press, 1971 ); D. Gross and S.B. Trelman, Phys. Rev. D4 (1971) 1059.
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