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Effects of compositeness in deep inelastic scattering
Volume 129B, number 5
PHYSICS LETTERS
29 September 1983
EFFECTS OF COMPOS1TENESS IN DEEP INELASTIC SCATTERING R. R/~ICKL Max-Planck-Institut ffir Physik und A strophysik, WernerHeisenberg lnstitut far Physik, D 8000 Mfinchen 40, Fed. Rep. Germany and Sektion Physik, Universit{#Mfinchen, D 8000 Mfinchen 2, Fed. Rep. Germany Received 26 April 1983 Revised manuscript received 24 June 1983
It is conjectured that leptons and quarks are composites of more fundamental constituents bound together by a new confining force with a typical confinement radius ~I/A H. Assuming that the effective low energy lagrangian is given by the lagrangian of the standard model plus induced lepton and quark current-current interactions, the effects of the latter remnants of the new force on the deep inelastic structure function, F2(x, Q2), are investigated at Q2 .~ Ah" It is shown that these signatures of compositeness should be detectable at HERA ifA H ~<2-3 TeV.
At presently accessible energies the dynamics o f leptons and hadrons is well described by the standard model of elementary leptons and quarks with fundamental gauge interactions based on the gauge group SU(3)c × (SU(2) × U(1))ew. The electroweak symmetry is spontaneously broken to the U(1)em o f electromagnetism at the empirical scale 1 / @ F ~ 300 GeV set b y the Fermi constant G F. Despite the impressive success of the standard theory it is commonly believed that it cannot be the final answer. The increasing number o f lepton and quark species one is observing as well as other unexplained features o f the model may in fact indicate *1 that leptons and quarks are not of elementary nature but are composites of more fundamental constituents ( " p r e o n s " ) bound together by a new force ("hypercolour force"). This force should probably be confining and become strong at a scale A H which is also a measure for the typical size of preon b o u n d states. Direct experimental tests of this new structure require energies of the order o f A H. Such tests may, therefore, be feasible, if at all, only in the far future. More important is to investigate effects of compositeness which are already detectable at energies consid,1 For a
recent discussion and more references see e.g. ref.
111. 0 . 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
erably below A H. A rather comprehensive discussion o f possible signatures can be found, for example, in ref. [ 1 ]. Generally, the low energy effects originate in differences between the effective low energy lagrangian o f the truly fundamental theory and the lagrangian of the standard model. These differences are presumably rather small and, thus, it is not surprising that there is no positive evidence from experiments so far. However, there exist bounds on A H [1]. Those which are relatively model-independent (e.g. from g - 2 measurements) constrain the compositeness scale to A H > 0 (0.5 TeV). Others [e.g. from ;t ~ e7 and AM(K~ -- KO)], although model-dependent, suggest even A H > 0 (100 TeV). These bounds provide some indications that A H is distinctly larger than the weak scale 1/x/-GF ~ 0_3 TeV. On the other hand, A H O(G~-1/2) can certainly not be excluded. Both of the above possibilities are discussed in the literature [1,2] in the context o f specific composite models. A H >~ GF 1/2 is natural in a scheme where all electroweak gauge bosons, the p h o t o n as well as the W-* and Z 0, are elementary while the fermions possess a new substructure at the scale A H. On the other hand, if one considers the weak interactions as residual interactions induced by the fundamental hypercolor force the weak bosons are also composite and one, obviously, would take A H ~ O(G~I/2). The proponents of b o t h 363
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schemes, however, probably agree that the lagrangian relevant at the scale G~ 1/2 should not be drastically different from the lagrangian of the standard model. This is easy to achieve in the first class of models simply by making A H sufficiently big. In models which interpret weak interactions as residual interactions one, for example, invokes [3] "W-dominance" and a large mass splitting between the ground state bosons and the excited weak bosons. Phenomenologically it is, therefore, reasonable [1, 2,4] to approximate the low energy effective lagrangian, "~eff, by the lagrangian of the standard model plus higher dimensional operators (e.g. 4-fermion operators) involving the fields of the standard model (and eventually some Goldstone fields). Thus, one has, schematically [4], d2eff = "~ standard + (g 2 / A 2 )
Oilkl t)it~j~k ~l + ....
(1)
Here, gH describes the effective strength of the nonstandard interactions and A H characterizes their typical energy scale. The fields, fin, are helicity components of the composite fermion fields (quarks and leptons). The Lorentz and colour structure which depends on details of the composite model one is considering is suppressed in eq. (1). It is also important to notice that not all flavour and helicity combinations must appear in "~eff with equal strength [1 ]. Some of them can be absent or, at least, suppressed due to selection rules enforced by symmetries of the preon dynamics. This possibility is taken into account in eq. (1) by the coefficients Oifkl. Depending on the particular picture of compositeness one has in mind one makes different assumptions on the parameters gH and A H. Referring to the two schemes addressed above one would consider g2H/4rr~ O(1) and A H >> G{ 1/2 on the supposition that the weak interactions are fundamental and the preon binding force is strong at distances ~ I / A H . In contrast, if the weak interactions are residual interactions one would associate gH with the weak coupling constant g and substitute for A H a typical mass, M', of the excited weak bosons. Thus, as long as Q2 ,~ A 2 or M '2 both possibilities can be investigated simultaneously using eq. (1). For definiteness, I shall focus on the case A H > GF 1/2 andg2/4rt = 1 and remark on the alternative parametrization mentioned above at the end of this paper. Clearly, the presence of non-standard interactions in ./2ef f will affect, in general, the values of the weak parameters, 364
29 September 1983
e.g. G F and sin20w, which appear in Z?standard and which are usually determined from very low energy (<'~ AN) experiments disregarding any modifications of the standard theory. As an immediate result, some of the standard model relations among the various weak parameters would be disturbed, a very interesting signature which should be searched for. These effects, however, will be small provided A N is sufficiently large as I shall assume in what follows. For illustration, an order of magnitude condition is obtained by requiring 4GF/X/~>g2/A 2. Taking g2/41r = l this corresponds to A H > 0.6 TeV. Recently, Eichten et al. [4] have used the effective lagrangian, eq. (1), to obtain limits on A u from the cross sections, do/d cos 0, for e+e - ~ e+e - and e+e -~/a+/a - measured at PETRA. More specifically, these authors have investigated deviations from the standard model results due to contributions from effective 4fermion operators in "~eff, eq. (1), of the form
++.(g2H/2A2)(~'~ue)h(-~'~ue)h
(2)
and +(g2H / 6 2) (~Tu e)h(t77 ut0 h .
(3)
Several possible products of vector (V) and axialvector (A) currents, namely h = V, A, V - A and V + A have been considered. Forg2H/4rr = 1 the following bounds on A H have been derived: AH(e+e - ~ e+e - ) > 0.75 TeV,
for (V -T-A) X (V ~- A ) ,
> 1.5 TeV,
forVXV, A× A
(4)
and AH(e+e - ~ p + p - ) > 1.4 TeV,
for (V -7-A)e × (V -7-A)u ,
> 2.2 TeV,
for V e × Vu, A e × Au .
(5)
In this letter, I discuss the effects of an eventual substructure of leptons and quarks on the total cross sections of deep inelastic electron-proton scattering in a similar dynamical framework. In particular, I shall investigate the sensitivity to A H of an ep-collider such as HERA which is capable to reach momentum transfers, Q2, of the order of a few 104 GeV 2 with reasonable
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statistics [5]. To anticipate a typical result, at Q2 104 GeV 2 one should be able to set a lower limit of A H > 1.5 TeV increasing to A H > 2.5 TeV at Q2 ~ 3 × 104 GeV 2. Furthermore, if there is a positive effect at Q2 ~ 104 GeV 2 accurate data (errors < 30%) would even provide information on the Lorentz structure of the effective lepton and quark currents involved in the induced current-current interactions in "~eff, eq. (1), and thus, probe details of the underlying preon dynamics. The basic presupposition for an effect of the kind described above to exist is that electrons and light quarks (u, d) have constituents in common [4]. If this is the case one would expect induced 4-fermion operators of the form
+-(g2/A2Hl('67Uelhe(YqTuqlhq ,
(6)
in the effective lagrangian of eq. (1) which could significantly change the standard model expectations on deep inelastic cross sections. In general, there could also exist scalar or tensor interactions. Their strength, however, depends crucially on the chiral properties of the preon theory. The reason why I am particularly considering vector and axial vector interactions is simply that the latter conserve helicity and are, therefore, not suppressed by a chiral symmetry at the preon level. To illustrate the sensitivity of the modifications to the helicity structure of the induced interaction, eq. (6), I consider the four current-current products with h e = (V T- A)e and hq = (V ~- A)q, separately. Using eqs. (1) and (6) the full amplitude for elastic eq-scattering is given by
M=(e2QeQq/qZ)[ff(p'e)~/UU(pe)] [~l(pq).,t#U(pq)] +
29 September 1983
eters in eq. (7) are mostly self-explanatory. The neutral current couplings are given by Ve=--~ +sin20w,
ae=l,
Vq =itql -Qqsin20w ,
aq =½tq ,
(8)
with sin20 w = 0.229 + 0.010 [6] and tq denoting the third component of the weak isospin of a q-type quark. For the Z-boson mass I use the value M Z = 90 GeV. s e and Sq take the values -+1 depending on the (V, A)structure one assumes in eq. (6), A straightforward calculation leads, then, to the following differential cross section (Q2 = Iq21 = 2p e.pe;y =q "Pq/PePq; s = sin 0w, t
c = cos 0w):
22 47ro~2QeQq
do -(eq ~ eq) dq2
Q4
× {Fq(Q2)[1 +(1
-y)21/2
+ Gq(Q2)[1 - (1
_y)2]/2},
(9)
where Fq(Q 2) = 1 +
2oevq
Q2
QeQqc2S2 QZ +M 2 Q2
]2
+(Oe(QeQqC2S2) + ae )(o +aq 2 ) (e27Mz 1 +Rq(e2)' 2aea q Gq (Qz)=QeQqczs 2 4Oeae'vqaq
Q2
Q2 +M 2 Q2
2
+(QeQqc2S2) 2 (Q~+M2) +SeSqRq(Q 2)
(lO)
e2 and
cos20 w sin20w(q 2 - M 2)
Rq(Q 2)
× [g(p'e)TU(Oe - ae75)U(Pe)] X [17(p'q~/,(Vq - aq75)U(pq)]
×
+ (gZ/aA~) [~(p'e)TU(1 - Se~5)U(Pe)] X [ff(p'q)')'u(1 -- SqT5)U(pq)] .
+
8.o eQq
1
)
I_ 1 _ (oe + Seae)(Oq + Sqaq) [ Q2 QeQqcZs2 ~ Q2 +M 2 ! g2 ~ Q2
(7)
The individual contributions to the above amplitude come from photon and Z 0 exchange and from the nonstandard interaction, eq. (6), respectively. The param-
It is the coefficient, Rq(Q2), by which the cross section, eq. (9), differs from the corresponding standard 365
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model result. The first two terms ofRq(Q2), eq. (11), come from the interference between the induced interaction, eq. (6), and the photon and the Z 0 exchange, respectively, the third one is the contribution to the cross section from the nonstandard interaction alone. The alternating signs in eq. (11) correspond to the sign one has in eq. (6). From the elastic electron-quark cross section, eq. (9), one obtains the total ep-cross section by folding eq. (9) with the quark distribution function, q(~, Q2), of a given quark species and summing over all quark and antiquark constituents of the proton. It is, furthermore, convenient to normalize the result such as to obtain the usual structure function, F2(x, Q2) = ZfQ2xqf(x, Q2), if only the one-photon exchange contributes, that is
do; ,/4rra2Q2s 1 + (1 _ y ) 2 F2(x ' Q2) = d-~-yy/
-Q~
~
2
'
(12)
where x/sis the CM energy and x = Q2/y .s. Explicitly, one fmds,
F2(X ' Q2) = ~ f Q2fxqf(x ' Q2) X ( F f ( Q 2) +
SGf(Q 2) 1 - (1 - y)2 ~
1V -y)2 f"
(13)
The coefficients.Ff and Gf are given in eq. (10) and S = + 1 ( - 1 ) for quark (antiquark) contributions. For the numerical computation o f F 2 I use the quark distributions of ref. [7] which take into account scaling violations in accordance with the QCD evolution in leading log-approximation and which are consistent with in. elastic data in the presently accessible Q2 range. Of course, it is somewhat dangerous to blindly extrapolate such numerical solutions of the QCD evolution equations to asymptotic Q2 ~ (104_105 GeV2). On the other hand, the ambiguities from this extrapolation mainly affect the absolute normalization of F2(x ' Q2) and have completely negligible influence on the deviations o f F 2 from its standard model behaviour which our interest is focussing on. I have checked the above assertion varying AQCD from 100 MeV to 500 MeV. At Q2 > 100 GeV 2 I find practically no variation o f F 2 at small x (x "- 0.1) and a dedecrease of the absolute normalization of 50% at most at large x (x ~ 0.9). The generation of a heavy quark 366
29 September 1983
100
e~u~e+
! [F(021*--GIQ2)]:
u
AH=
1TeV
1
....
10 2
10 3
10~
I0s
o2(GeV21 Fig. 1. The coefficients F(Q2) +_G(Q:) for elastic electron(u)quark scattering and various induced current-current interactions: (1) (F + G) for + (V - A)e X (V - A)u, (2) (F +G) for - (V - A)e X (V - A)u , (3) (F - G) for +(V - A)e × (V - A)u ,(1') (F - G) for +(V - A)e X (V+ A)u , (2') (F - G) for -(V - A)e × (V + A)u, (3') (F + G) for +(V - A)e × (V + A)u. The standard model predictions are labelled by (F +-G)ST.
component (c, b, ...) in the nucleon sea via QCD evolution is also neglected, a simplification which has no effect on the conclusions of this letter. The deviation of the coefficients Fu(Q 2) and Gu(Q2), eq. (10), from their standard model behaviour in Q2 is illustrated in fig. 1 for atypical scale, A H = 1 TeV, and various assumptions on the Lorentz structure of the induced interactions. It is actually simpler to consider the functions, F + G, because in some of these combinations the nonstandard contributions cancel. To wit (F + G) [+(V - A)e X (V + A)q; +(V + A)e × (V - A ) q ] = (F + G)S T , (F - G) [+(V - A)e X (V - A)q; +-(V + A)e X (V + A)q] = (F - G)S T
(14)
as can be easily seen from eq. (10). Furthermore, one has the approximate equalities
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(F + G)[+(V - A)e X (V -- A)q] ~-- (/7 + G) [-+(V + A)e X (V + A)q ] , (F - C)[+(V - A)e X (V ~-- (F
-
A)q]
+
G)[+(V + A)e X (V - A)q] ,
(15)
which hold within 10 to 20%. The remaining independent coefficients are shown in fig. 1. The most important observation is that one expects sizable ( > factor of 2) deviations from the standard model coefficients already at Q2 > 4000 GeV 2, in other words, far below A2H = 106 GeV 2. Also interesting is the sensitivity of the effects due to the current-current interaction, eq. (6), to its overall sign relative to the standard contributions as well as to its Lorentz structure. For a d-type quark one obtains a similar result except that the influence of the sign in eq. (6) on the pattern shown in fig. 1 is reversed and that, on the whole, the coefficients F d and G d differ even more from the standard model values, namely by a factor 2 at Q2 ~ 2000 GeV 2 . Next, fig. 2 exhibits how the residual interaction, eq. (6), induced by the fundamental hypercolour force would reveal itself in the structure function F2(x , Q2). Some features of the behaviour o f f 2 follow directly
29 September 1983
from eq. (13) and the properties of the coefficients F and G, eq. (10), shown in fig. 1 : (i) (V + A)e X (V + A)q and (V + A)e X (V - A)q current-current interactions cannot be distinguished from (V - a)e X (V - A)q and (V - A)e X (V + n)q interactions, respectively. This is an immediate consequence of eqs. (14) and (15). (ii) At fixed x and small Q2, i.e. smally, also ( v - A)e × (V - A)q and (V - A)e X (W + A)q contributions lead to rather similar modifications. This follows from the fact that F(Q 2) is rather insensitive to the helicity structure of the currents and that G(Q2), which is very sensitive (it even flips the sign), is multiplied in eq. (13) by a function o f y which vanishes at y = 0. At fixed x and large Q2, on the other hand, it is precisely the negative sign of G(Q 2) in the (V - A)e × (V + A)q case which leads to a suppression o f f 2 relative to the structure function expected for (V - A)e × (V - A)q. Concerning the prospects of detecting such effects of an eventual lepton and quark substructure, fig. 2 assures that one should have already observable signatures at Q2 < 104 GeV 2 if the nonstandard term, eq. (6), in ~ e f f comes with a posi-
F2 (x'Q2);C'~'=314GeV
I
/
,oo
F2(x: 0.5,Q2);I/s'=314GeV / AH=I;
1.0 0.5
°.'
.........
0.05 i
10 3
=
i
I
J
iJll
5.10 3 10 4
i
J
i
5.10 4
Q2(GeV2) Fig. 2. The structure function F2(x, Q2) versus Q2 at fixed x = 0.5 for various induced current-current interactions: (1) +(V - A)e X (V - A)q, (1') -(V - A)e X (V - A)q, (2) +(V - A) e X (V + A)q, (2') -(V - A)e X (V + A)q. The standard model prediction is labelled by ST.
10 2
10 3
10 4
10 5
Q2(GeV2) Fig. 3. The structure function F2(x, Q2) versus Q2 for various values ofx. The full (dashed) curves illustrate the effect of a nonstandard +(V - A)e X (V - A)q [+(V - A)e × (V + A)q] interaction whereas the dotted curves illustrate the standard model predictions. 367
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tive sign, whereas at Q2 ~ 104 GeV 2 one should see a clear deviation from the standard model for all cases considered, in partiucular however, for an induced (V - A)e X (V - A)q interaction. How the effects discussed above depend on x is illustrated in fig. 3. At Q2 <~ 2 X 104 GeV 2 there is only little variation of the overall pattern with x. The intention of fig. 4, finally, is to indicate the bounds on A H one could expect from a ep-collider like HERA. The Q2-behaviour of the structure function F2(x ' Q2), eq. (13), at x = 0.5 and for various values of A H ranging from 0.75 TeV to 3 TeV (solid curves) is compared to the standard model result (dashed curve). Also shown is the structure function obtained from one-photon exchange only (dotted curve). We see that the standard neutral current interaction leads to an increase o f f 2 relative to the pure electromagnetic form factor by about 25% at Q2 ~ 104 GeV 2 and by as much as a factor of 2 at Q2 ~ 3 × 104 GeV 2. An effect of this magnitude is certainly detectable. If there exists an effective current--current interaction of the form assumed in eq. (6) on top of the Z 0, F 2 ( x ' Q2) is predicted to increase even faster with Q2
AH{TeV) 5.0
/0.75
F2( x =0.5,Q2);l/E=31/,GeV /
/1.0 ~ /
1.0
0.05
.......
::..z.-..7 -~
. . . . . . . . . . . . . . . . 1-y I
10 3
I
I
I
i llil
5.10 3 10 ~
i
i
i
I
i
5-10 ~
0,2 ( GeV 2)
Fig. 4. The structure function F2(x, Q2) versus Q2 at fixed x = 0.5 for various scales AH controlling the strength of an induced +(V - A)e X (V - A)q current-current interaction. The dashed and dotted curves show the standard model prediction and the pure one-photon exchange result, respectively. 368
29 September 1983
at fixed x. This is illustrated in fig. 4 for the case of a (V - A)e X (V - A)o interaction and a positive overall sign in eq. (6). The size of the deviation from the standard model can be characterized as follows. At Q2 = 104 GeV 2 (x > 0.1). A F 2 / F 2 ~ 25% ,
60%,
forA H~2TeV, for A H ~ 1.5 T e V ,
(16)
whereas at Q2 ___3 × 104 GeV 2 (x > 0.3) AF2/F 2 ~130% ,
3 5%,
forA H~2TeV, for A H ~ 3 TeV.
(17)
Thus, the absence of an effect at the level of 30% would give a bound A H > 2 - 3 TeV. For other possible Lorentz structures of the induced interaction, eq. (6), the effects are somewhat less pronounced (see fig. 2) and, therefore, the obtainable bounds somewhat weaker (A H > 1.5-2 TeV). With appropriate changes the present analysis is applicable to a variety of specific composite models [2, 3] as pointed out earlier. In particular, in models where also the weak bosons are composite one expects excited W' and Z!-bosons which at Q2 < M ' 2 would mimick local current-current interactions of the kind proposed in eq. (6). Considering this case the above bounds on A H can be translated in bounds on the mass, M', provided the contributions from the groundstate W's and Z are effectively described by the standard model not only at very low Q2 but also at Q2 O(M 2 z)" The coefficient g 2 / A 2 in eq. (6) is, then, simply re'placed by g'2/2M'2 where g' is the coupling of the Z' to lepton and quark currents. Taking for g' the value of the weak coupling g of the standard model, g'2/4rt = g2/4zr ~ 0.04, the bound A H > 2 TeV corresponds t o M ' > 300 GeV. Note, however, that the results plotted in figs. 1-3 and in fig. 4 for A H ~< 2 TeV would change quantitatively due to propagator effects which are neglected in the present analysis but become important as Q2 approaches M '2. Basically, the magnitude of the deviations from the standard model exhibited in figs. 1-3 would be reduced by roughly a factor 2 at Q2 ~ 3 X 104 GeV 2 . Their qualitative pattern, however, would still persist. To conclude, I have investigated effects of an eventual quark and lepton substructure on the structure function, F2(x , Q2), in deep inelastic ep scattering. In
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particular, I have assumed (i) that quarks and leptons are (almost) massless composites of constituents bound together by a new confining force with a typical confinement radius ~ I / A H , (ii) that A H is considerably larger than the weak scale (GF) -1/2 ~ 300 GeV, and (iii) that at Q2 ~ A 2 the effective lagrangian is given by the lagrangian of the standard model up to four-fermion operators with coefficients "~1/A 2 , leading to nonstandard lepton and quark current-current interactions in cases where some of the preons involved are identical. These low energy remnants of the hypercolor force have considerable effects on F2(x ' Q2) which are shown to become detectable at Q2 ~ O(104 GeV 2) if A t / ~ 2 TeV. Thus, the prospects of obtaining information on compositeness from an ep-collider like HERA look promising. It should be made totally clear, however, that this large value for A H is obtained assuming g2/47r = 1 for the effective coupling constant of the non-standard interactions. More signatures of substructure in charge and polarization asymmetries as well as in the charged current process ep ~ vX are under investigation.
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I would like to thank A. Buras, H. Fritzsch, D. Schildknecht and, in particular, R.D. Peccei for useful discussions.
References [1 ] H. Harari, Composite models for quarks and leptons, Weizmann Institute preprint, WIS-82/60 Dec-Ph (Dec. 1982); R.D. Peccei, Composite models of quarks and leptons, MPI-preprint, MPI-PAE/PTh 69/82 (Oct. 1982). [2] See discussion by M.E. Peskin, Compositeness of quarks and leptons, Proc. Intern. Symp. on Lepton and photon interactions at high energies (Bonn, 1981), ed. W. Pfeil (Phys. Inst. Univ. Bonn, 1981) p. 880. [3] H. Fritzsch, D. Schildknecht and R. K6gerler, Phys. Lett. 114B (1982) 157; M. Kuroda and D. Schildknecht, Phys. Lett. 121B (1983) 173. [4] E.J. Eichten, K.D. Lane and M.E. Peskin, Phys. Rev. Lett. 50 (1983) 811. [51 B.H. Wiik, HERA-Report, Resum6 of the Discussion Meeting Physics with ep colliders, Wuppertal, Oct. 2-3, 1981, DESY HERA 81118 (Oct. 1981). [6] J.E. Kim, P. Langacker, M. Levine and H.H. Williams, Rev. Mod. Phys. 53 (1981) 211. [7] R. Baler, J. Engels and B. Petersson, Z. Phys. C2 (1979) 265.
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EFFECTS OF COMPOS1TENESS IN DEEP INELASTIC SCATTERING R. R/~ICKL Max-Planck-Institut ffir Physik und A strophysik, WernerHeisenberg lnstitut far Physik, D 8000 Mfinchen 40, Fed. Rep. Germany and Sektion Physik, Universit{#Mfinchen, D 8000 Mfinchen 2, Fed. Rep. Germany Received 26 April 1983 Revised manuscript received 24 June 1983
It is conjectured that leptons and quarks are composites of more fundamental constituents bound together by a new confining force with a typical confinement radius ~I/A H. Assuming that the effective low energy lagrangian is given by the lagrangian of the standard model plus induced lepton and quark current-current interactions, the effects of the latter remnants of the new force on the deep inelastic structure function, F2(x, Q2), are investigated at Q2 .~ Ah" It is shown that these signatures of compositeness should be detectable at HERA ifA H ~<2-3 TeV.
At presently accessible energies the dynamics o f leptons and hadrons is well described by the standard model of elementary leptons and quarks with fundamental gauge interactions based on the gauge group SU(3)c × (SU(2) × U(1))ew. The electroweak symmetry is spontaneously broken to the U(1)em o f electromagnetism at the empirical scale 1 / @ F ~ 300 GeV set b y the Fermi constant G F. Despite the impressive success of the standard theory it is commonly believed that it cannot be the final answer. The increasing number o f lepton and quark species one is observing as well as other unexplained features o f the model may in fact indicate *1 that leptons and quarks are not of elementary nature but are composites of more fundamental constituents ( " p r e o n s " ) bound together by a new force ("hypercolour force"). This force should probably be confining and become strong at a scale A H which is also a measure for the typical size of preon b o u n d states. Direct experimental tests of this new structure require energies of the order o f A H. Such tests may, therefore, be feasible, if at all, only in the far future. More important is to investigate effects of compositeness which are already detectable at energies consid,1 For a
recent discussion and more references see e.g. ref.
111. 0 . 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
erably below A H. A rather comprehensive discussion o f possible signatures can be found, for example, in ref. [ 1 ]. Generally, the low energy effects originate in differences between the effective low energy lagrangian o f the truly fundamental theory and the lagrangian of the standard model. These differences are presumably rather small and, thus, it is not surprising that there is no positive evidence from experiments so far. However, there exist bounds on A H [1]. Those which are relatively model-independent (e.g. from g - 2 measurements) constrain the compositeness scale to A H > 0 (0.5 TeV). Others [e.g. from ;t ~ e7 and AM(K~ -- KO)], although model-dependent, suggest even A H > 0 (100 TeV). These bounds provide some indications that A H is distinctly larger than the weak scale 1/x/-GF ~ 0_3 TeV. On the other hand, A H O(G~-1/2) can certainly not be excluded. Both of the above possibilities are discussed in the literature [1,2] in the context o f specific composite models. A H >~ GF 1/2 is natural in a scheme where all electroweak gauge bosons, the p h o t o n as well as the W-* and Z 0, are elementary while the fermions possess a new substructure at the scale A H. On the other hand, if one considers the weak interactions as residual interactions induced by the fundamental hypercolor force the weak bosons are also composite and one, obviously, would take A H ~ O(G~I/2). The proponents of b o t h 363
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schemes, however, probably agree that the lagrangian relevant at the scale G~ 1/2 should not be drastically different from the lagrangian of the standard model. This is easy to achieve in the first class of models simply by making A H sufficiently big. In models which interpret weak interactions as residual interactions one, for example, invokes [3] "W-dominance" and a large mass splitting between the ground state bosons and the excited weak bosons. Phenomenologically it is, therefore, reasonable [1, 2,4] to approximate the low energy effective lagrangian, "~eff, by the lagrangian of the standard model plus higher dimensional operators (e.g. 4-fermion operators) involving the fields of the standard model (and eventually some Goldstone fields). Thus, one has, schematically [4], d2eff = "~ standard + (g 2 / A 2 )
Oilkl t)it~j~k ~l + ....
(1)
Here, gH describes the effective strength of the nonstandard interactions and A H characterizes their typical energy scale. The fields, fin, are helicity components of the composite fermion fields (quarks and leptons). The Lorentz and colour structure which depends on details of the composite model one is considering is suppressed in eq. (1). It is also important to notice that not all flavour and helicity combinations must appear in "~eff with equal strength [1 ]. Some of them can be absent or, at least, suppressed due to selection rules enforced by symmetries of the preon dynamics. This possibility is taken into account in eq. (1) by the coefficients Oifkl. Depending on the particular picture of compositeness one has in mind one makes different assumptions on the parameters gH and A H. Referring to the two schemes addressed above one would consider g2H/4rr~ O(1) and A H >> G{ 1/2 on the supposition that the weak interactions are fundamental and the preon binding force is strong at distances ~ I / A H . In contrast, if the weak interactions are residual interactions one would associate gH with the weak coupling constant g and substitute for A H a typical mass, M', of the excited weak bosons. Thus, as long as Q2 ,~ A 2 or M '2 both possibilities can be investigated simultaneously using eq. (1). For definiteness, I shall focus on the case A H > GF 1/2 andg2/4rt = 1 and remark on the alternative parametrization mentioned above at the end of this paper. Clearly, the presence of non-standard interactions in ./2ef f will affect, in general, the values of the weak parameters, 364
29 September 1983
e.g. G F and sin20w, which appear in Z?standard and which are usually determined from very low energy (<'~ AN) experiments disregarding any modifications of the standard theory. As an immediate result, some of the standard model relations among the various weak parameters would be disturbed, a very interesting signature which should be searched for. These effects, however, will be small provided A N is sufficiently large as I shall assume in what follows. For illustration, an order of magnitude condition is obtained by requiring 4GF/X/~>g2/A 2. Taking g2/41r = l this corresponds to A H > 0.6 TeV. Recently, Eichten et al. [4] have used the effective lagrangian, eq. (1), to obtain limits on A u from the cross sections, do/d cos 0, for e+e - ~ e+e - and e+e -~/a+/a - measured at PETRA. More specifically, these authors have investigated deviations from the standard model results due to contributions from effective 4fermion operators in "~eff, eq. (1), of the form
++.(g2H/2A2)(~'~ue)h(-~'~ue)h
(2)
and +(g2H / 6 2) (~Tu e)h(t77 ut0 h .
(3)
Several possible products of vector (V) and axialvector (A) currents, namely h = V, A, V - A and V + A have been considered. Forg2H/4rr = 1 the following bounds on A H have been derived: AH(e+e - ~ e+e - ) > 0.75 TeV,
for (V -T-A) X (V ~- A ) ,
> 1.5 TeV,
forVXV, A× A
(4)
and AH(e+e - ~ p + p - ) > 1.4 TeV,
for (V -7-A)e × (V -7-A)u ,
> 2.2 TeV,
for V e × Vu, A e × Au .
(5)
In this letter, I discuss the effects of an eventual substructure of leptons and quarks on the total cross sections of deep inelastic electron-proton scattering in a similar dynamical framework. In particular, I shall investigate the sensitivity to A H of an ep-collider such as HERA which is capable to reach momentum transfers, Q2, of the order of a few 104 GeV 2 with reasonable
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statistics [5]. To anticipate a typical result, at Q2 104 GeV 2 one should be able to set a lower limit of A H > 1.5 TeV increasing to A H > 2.5 TeV at Q2 ~ 3 × 104 GeV 2. Furthermore, if there is a positive effect at Q2 ~ 104 GeV 2 accurate data (errors < 30%) would even provide information on the Lorentz structure of the effective lepton and quark currents involved in the induced current-current interactions in "~eff, eq. (1), and thus, probe details of the underlying preon dynamics. The basic presupposition for an effect of the kind described above to exist is that electrons and light quarks (u, d) have constituents in common [4]. If this is the case one would expect induced 4-fermion operators of the form
+-(g2/A2Hl('67Uelhe(YqTuqlhq ,
(6)
in the effective lagrangian of eq. (1) which could significantly change the standard model expectations on deep inelastic cross sections. In general, there could also exist scalar or tensor interactions. Their strength, however, depends crucially on the chiral properties of the preon theory. The reason why I am particularly considering vector and axial vector interactions is simply that the latter conserve helicity and are, therefore, not suppressed by a chiral symmetry at the preon level. To illustrate the sensitivity of the modifications to the helicity structure of the induced interaction, eq. (6), I consider the four current-current products with h e = (V T- A)e and hq = (V ~- A)q, separately. Using eqs. (1) and (6) the full amplitude for elastic eq-scattering is given by
M=(e2QeQq/qZ)[ff(p'e)~/UU(pe)] [~l(pq).,t#U(pq)] +
29 September 1983
eters in eq. (7) are mostly self-explanatory. The neutral current couplings are given by Ve=--~ +sin20w,
ae=l,
Vq =itql -Qqsin20w ,
aq =½tq ,
(8)
with sin20 w = 0.229 + 0.010 [6] and tq denoting the third component of the weak isospin of a q-type quark. For the Z-boson mass I use the value M Z = 90 GeV. s e and Sq take the values -+1 depending on the (V, A)structure one assumes in eq. (6), A straightforward calculation leads, then, to the following differential cross section (Q2 = Iq21 = 2p e.pe;y =q "Pq/PePq; s = sin 0w, t
c = cos 0w):
22 47ro~2QeQq
do -(eq ~ eq) dq2
Q4
× {Fq(Q2)[1 +(1
-y)21/2
+ Gq(Q2)[1 - (1
_y)2]/2},
(9)
where Fq(Q 2) = 1 +
2oevq
Q2
QeQqc2S2 QZ +M 2 Q2
]2
+(Oe(QeQqC2S2) + ae )(o +aq 2 ) (e27Mz 1 +Rq(e2)' 2aea q Gq (Qz)=QeQqczs 2 4Oeae'vqaq
Q2
Q2 +M 2 Q2
2
+(QeQqc2S2) 2 (Q~+M2) +SeSqRq(Q 2)
(lO)
e2 and
cos20 w sin20w(q 2 - M 2)
Rq(Q 2)
× [g(p'e)TU(Oe - ae75)U(Pe)] X [17(p'q~/,(Vq - aq75)U(pq)]
×
+ (gZ/aA~) [~(p'e)TU(1 - Se~5)U(Pe)] X [ff(p'q)')'u(1 -- SqT5)U(pq)] .
+
8.o eQq
1
)
I_ 1 _ (oe + Seae)(Oq + Sqaq) [ Q2 QeQqcZs2 ~ Q2 +M 2 ! g2 ~ Q2
(7)
The individual contributions to the above amplitude come from photon and Z 0 exchange and from the nonstandard interaction, eq. (6), respectively. The param-
It is the coefficient, Rq(Q2), by which the cross section, eq. (9), differs from the corresponding standard 365
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model result. The first two terms ofRq(Q2), eq. (11), come from the interference between the induced interaction, eq. (6), and the photon and the Z 0 exchange, respectively, the third one is the contribution to the cross section from the nonstandard interaction alone. The alternating signs in eq. (11) correspond to the sign one has in eq. (6). From the elastic electron-quark cross section, eq. (9), one obtains the total ep-cross section by folding eq. (9) with the quark distribution function, q(~, Q2), of a given quark species and summing over all quark and antiquark constituents of the proton. It is, furthermore, convenient to normalize the result such as to obtain the usual structure function, F2(x, Q2) = ZfQ2xqf(x, Q2), if only the one-photon exchange contributes, that is
do; ,/4rra2Q2s 1 + (1 _ y ) 2 F2(x ' Q2) = d-~-yy/
-Q~
~
2
'
(12)
where x/sis the CM energy and x = Q2/y .s. Explicitly, one fmds,
F2(X ' Q2) = ~ f Q2fxqf(x ' Q2) X ( F f ( Q 2) +
SGf(Q 2) 1 - (1 - y)2 ~
1V -y)2 f"
(13)
The coefficients.Ff and Gf are given in eq. (10) and S = + 1 ( - 1 ) for quark (antiquark) contributions. For the numerical computation o f F 2 I use the quark distributions of ref. [7] which take into account scaling violations in accordance with the QCD evolution in leading log-approximation and which are consistent with in. elastic data in the presently accessible Q2 range. Of course, it is somewhat dangerous to blindly extrapolate such numerical solutions of the QCD evolution equations to asymptotic Q2 ~ (104_105 GeV2). On the other hand, the ambiguities from this extrapolation mainly affect the absolute normalization of F2(x ' Q2) and have completely negligible influence on the deviations o f F 2 from its standard model behaviour which our interest is focussing on. I have checked the above assertion varying AQCD from 100 MeV to 500 MeV. At Q2 > 100 GeV 2 I find practically no variation o f F 2 at small x (x "- 0.1) and a dedecrease of the absolute normalization of 50% at most at large x (x ~ 0.9). The generation of a heavy quark 366
29 September 1983
100
e~u~e+
! [F(021*--GIQ2)]:
u
AH=
1TeV
1
....
10 2
10 3
10~
I0s
o2(GeV21 Fig. 1. The coefficients F(Q2) +_G(Q:) for elastic electron(u)quark scattering and various induced current-current interactions: (1) (F + G) for + (V - A)e X (V - A)u, (2) (F +G) for - (V - A)e X (V - A)u , (3) (F - G) for +(V - A)e × (V - A)u ,(1') (F - G) for +(V - A)e X (V+ A)u , (2') (F - G) for -(V - A)e × (V + A)u, (3') (F + G) for +(V - A)e × (V + A)u. The standard model predictions are labelled by (F +-G)ST.
component (c, b, ...) in the nucleon sea via QCD evolution is also neglected, a simplification which has no effect on the conclusions of this letter. The deviation of the coefficients Fu(Q 2) and Gu(Q2), eq. (10), from their standard model behaviour in Q2 is illustrated in fig. 1 for atypical scale, A H = 1 TeV, and various assumptions on the Lorentz structure of the induced interactions. It is actually simpler to consider the functions, F + G, because in some of these combinations the nonstandard contributions cancel. To wit (F + G) [+(V - A)e X (V + A)q; +(V + A)e × (V - A ) q ] = (F + G)S T , (F - G) [+(V - A)e X (V - A)q; +-(V + A)e X (V + A)q] = (F - G)S T
(14)
as can be easily seen from eq. (10). Furthermore, one has the approximate equalities
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(F + G)[+(V - A)e X (V -- A)q] ~-- (/7 + G) [-+(V + A)e X (V + A)q ] , (F - C)[+(V - A)e X (V ~-- (F
-
A)q]
+
G)[+(V + A)e X (V - A)q] ,
(15)
which hold within 10 to 20%. The remaining independent coefficients are shown in fig. 1. The most important observation is that one expects sizable ( > factor of 2) deviations from the standard model coefficients already at Q2 > 4000 GeV 2, in other words, far below A2H = 106 GeV 2. Also interesting is the sensitivity of the effects due to the current-current interaction, eq. (6), to its overall sign relative to the standard contributions as well as to its Lorentz structure. For a d-type quark one obtains a similar result except that the influence of the sign in eq. (6) on the pattern shown in fig. 1 is reversed and that, on the whole, the coefficients F d and G d differ even more from the standard model values, namely by a factor 2 at Q2 ~ 2000 GeV 2 . Next, fig. 2 exhibits how the residual interaction, eq. (6), induced by the fundamental hypercolour force would reveal itself in the structure function F2(x , Q2). Some features of the behaviour o f f 2 follow directly
29 September 1983
from eq. (13) and the properties of the coefficients F and G, eq. (10), shown in fig. 1 : (i) (V + A)e X (V + A)q and (V + A)e X (V - A)q current-current interactions cannot be distinguished from (V - a)e X (V - A)q and (V - A)e X (V + n)q interactions, respectively. This is an immediate consequence of eqs. (14) and (15). (ii) At fixed x and small Q2, i.e. smally, also ( v - A)e × (V - A)q and (V - A)e X (W + A)q contributions lead to rather similar modifications. This follows from the fact that F(Q 2) is rather insensitive to the helicity structure of the currents and that G(Q2), which is very sensitive (it even flips the sign), is multiplied in eq. (13) by a function o f y which vanishes at y = 0. At fixed x and large Q2, on the other hand, it is precisely the negative sign of G(Q 2) in the (V - A)e × (V + A)q case which leads to a suppression o f f 2 relative to the structure function expected for (V - A)e × (V - A)q. Concerning the prospects of detecting such effects of an eventual lepton and quark substructure, fig. 2 assures that one should have already observable signatures at Q2 < 104 GeV 2 if the nonstandard term, eq. (6), in ~ e f f comes with a posi-
F2 (x'Q2);C'~'=314GeV
I
/
,oo
F2(x: 0.5,Q2);I/s'=314GeV / AH=I;
1.0 0.5
°.'
.........
0.05 i
10 3
=
i
I
J
iJll
5.10 3 10 4
i
J
i
5.10 4
Q2(GeV2) Fig. 2. The structure function F2(x, Q2) versus Q2 at fixed x = 0.5 for various induced current-current interactions: (1) +(V - A)e X (V - A)q, (1') -(V - A)e X (V - A)q, (2) +(V - A) e X (V + A)q, (2') -(V - A)e X (V + A)q. The standard model prediction is labelled by ST.
10 2
10 3
10 4
10 5
Q2(GeV2) Fig. 3. The structure function F2(x, Q2) versus Q2 for various values ofx. The full (dashed) curves illustrate the effect of a nonstandard +(V - A)e X (V - A)q [+(V - A)e × (V + A)q] interaction whereas the dotted curves illustrate the standard model predictions. 367
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tive sign, whereas at Q2 ~ 104 GeV 2 one should see a clear deviation from the standard model for all cases considered, in partiucular however, for an induced (V - A)e X (V - A)q interaction. How the effects discussed above depend on x is illustrated in fig. 3. At Q2 <~ 2 X 104 GeV 2 there is only little variation of the overall pattern with x. The intention of fig. 4, finally, is to indicate the bounds on A H one could expect from a ep-collider like HERA. The Q2-behaviour of the structure function F2(x ' Q2), eq. (13), at x = 0.5 and for various values of A H ranging from 0.75 TeV to 3 TeV (solid curves) is compared to the standard model result (dashed curve). Also shown is the structure function obtained from one-photon exchange only (dotted curve). We see that the standard neutral current interaction leads to an increase o f f 2 relative to the pure electromagnetic form factor by about 25% at Q2 ~ 104 GeV 2 and by as much as a factor of 2 at Q2 ~ 3 × 104 GeV 2. An effect of this magnitude is certainly detectable. If there exists an effective current--current interaction of the form assumed in eq. (6) on top of the Z 0, F 2 ( x ' Q2) is predicted to increase even faster with Q2
AH{TeV) 5.0
/0.75
F2( x =0.5,Q2);l/E=31/,GeV /
/1.0 ~ /
1.0
0.05
.......
::..z.-..7 -~
. . . . . . . . . . . . . . . . 1-y I
10 3
I
I
I
i llil
5.10 3 10 ~
i
i
i
I
i
5-10 ~
0,2 ( GeV 2)
Fig. 4. The structure function F2(x, Q2) versus Q2 at fixed x = 0.5 for various scales AH controlling the strength of an induced +(V - A)e X (V - A)q current-current interaction. The dashed and dotted curves show the standard model prediction and the pure one-photon exchange result, respectively. 368
29 September 1983
at fixed x. This is illustrated in fig. 4 for the case of a (V - A)e X (V - A)o interaction and a positive overall sign in eq. (6). The size of the deviation from the standard model can be characterized as follows. At Q2 = 104 GeV 2 (x > 0.1). A F 2 / F 2 ~ 25% ,
60%,
forA H~2TeV, for A H ~ 1.5 T e V ,
(16)
whereas at Q2 ___3 × 104 GeV 2 (x > 0.3) AF2/F 2 ~130% ,
3 5%,
forA H~2TeV, for A H ~ 3 TeV.
(17)
Thus, the absence of an effect at the level of 30% would give a bound A H > 2 - 3 TeV. For other possible Lorentz structures of the induced interaction, eq. (6), the effects are somewhat less pronounced (see fig. 2) and, therefore, the obtainable bounds somewhat weaker (A H > 1.5-2 TeV). With appropriate changes the present analysis is applicable to a variety of specific composite models [2, 3] as pointed out earlier. In particular, in models where also the weak bosons are composite one expects excited W' and Z!-bosons which at Q2 < M ' 2 would mimick local current-current interactions of the kind proposed in eq. (6). Considering this case the above bounds on A H can be translated in bounds on the mass, M', provided the contributions from the groundstate W's and Z are effectively described by the standard model not only at very low Q2 but also at Q2 O(M 2 z)" The coefficient g 2 / A 2 in eq. (6) is, then, simply re'placed by g'2/2M'2 where g' is the coupling of the Z' to lepton and quark currents. Taking for g' the value of the weak coupling g of the standard model, g'2/4rt = g2/4zr ~ 0.04, the bound A H > 2 TeV corresponds t o M ' > 300 GeV. Note, however, that the results plotted in figs. 1-3 and in fig. 4 for A H ~< 2 TeV would change quantitatively due to propagator effects which are neglected in the present analysis but become important as Q2 approaches M '2. Basically, the magnitude of the deviations from the standard model exhibited in figs. 1-3 would be reduced by roughly a factor 2 at Q2 ~ 3 X 104 GeV 2 . Their qualitative pattern, however, would still persist. To conclude, I have investigated effects of an eventual quark and lepton substructure on the structure function, F2(x , Q2), in deep inelastic ep scattering. In
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particular, I have assumed (i) that quarks and leptons are (almost) massless composites of constituents bound together by a new confining force with a typical confinement radius ~ I / A H , (ii) that A H is considerably larger than the weak scale (GF) -1/2 ~ 300 GeV, and (iii) that at Q2 ~ A 2 the effective lagrangian is given by the lagrangian of the standard model up to four-fermion operators with coefficients "~1/A 2 , leading to nonstandard lepton and quark current-current interactions in cases where some of the preons involved are identical. These low energy remnants of the hypercolor force have considerable effects on F2(x ' Q2) which are shown to become detectable at Q2 ~ O(104 GeV 2) if A t / ~ 2 TeV. Thus, the prospects of obtaining information on compositeness from an ep-collider like HERA look promising. It should be made totally clear, however, that this large value for A H is obtained assuming g2/47r = 1 for the effective coupling constant of the non-standard interactions. More signatures of substructure in charge and polarization asymmetries as well as in the charged current process ep ~ vX are under investigation.
29 September 1983
I would like to thank A. Buras, H. Fritzsch, D. Schildknecht and, in particular, R.D. Peccei for useful discussions.
References [1 ] H. Harari, Composite models for quarks and leptons, Weizmann Institute preprint, WIS-82/60 Dec-Ph (Dec. 1982); R.D. Peccei, Composite models of quarks and leptons, MPI-preprint, MPI-PAE/PTh 69/82 (Oct. 1982). [2] See discussion by M.E. Peskin, Compositeness of quarks and leptons, Proc. Intern. Symp. on Lepton and photon interactions at high energies (Bonn, 1981), ed. W. Pfeil (Phys. Inst. Univ. Bonn, 1981) p. 880. [3] H. Fritzsch, D. Schildknecht and R. K6gerler, Phys. Lett. 114B (1982) 157; M. Kuroda and D. Schildknecht, Phys. Lett. 121B (1983) 173. [4] E.J. Eichten, K.D. Lane and M.E. Peskin, Phys. Rev. Lett. 50 (1983) 811. [51 B.H. Wiik, HERA-Report, Resum6 of the Discussion Meeting Physics with ep colliders, Wuppertal, Oct. 2-3, 1981, DESY HERA 81118 (Oct. 1981). [6] J.E. Kim, P. Langacker, M. Levine and H.H. Williams, Rev. Mod. Phys. 53 (1981) 211. [7] R. Baler, J. Engels and B. Petersson, Z. Phys. C2 (1979) 265.
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