Event Shapes

A natural frame in which to study the dynamics of the hadronic final state in DIS is the Breit frame [22]. In this frame the exchanged virtual boson is purely space-like with 3-momentum ${\bf q}=(0,0,-Q)$, the incident quark carries momentum Q/2 in the positive Z direction, and the outgoing struck quark carries Q/2 in the negative Z direction. A final state particle has a 4-momentum p^B in this frame, and is assigned to the current region if p^B_Z is negative, and to the target frame if p^B_Z is positive. The advantage of this frame lies in the maximal separation of the outgoing parton from radiation associated with the incoming parton and the proton remnant, thus providing the optimal environment for the study of the fragmentation of the outgoing parton.

Event shape variables have been investigated in e⁺e^- experiments and used to extract the strong coupling constant $\alpha_s(M_Z)$ independent of any jet algorithm, see eg ref. [23]. H1 have recently performed a similar analysis [24] in deep inelastic scattering in the current fragmentation region of the Breit frame.

The event shape dependence on Q (or energy dependence) can be due to the logarithmic change of the strong coupling constant $\alpha_s(Q) \propto 1/\ln Q$, and/or power corrections (hadronisation effects) which are expected to behave like 1/Q. Recent theoretical developments suggest that 1/Q corrections are not necessarily related to hadronisation, but may instead be a universal soft gluon phenomenon associated with the behaviour of the running coupling at small momentum scales [25,26].

H1 have analysed a number of infrared safe (ie independent of the number of partons produced) event shape variables. Their definitions are given below, where the sums extend over all hadrons h (being a calorimetric cluster in the detector or a parton in the QCD calculations) with four-momentum $p^B_h = \{ E^B_h,\, {\bf p}^B_h \}$ The current hemisphere axis ${\bf n} = \{0,\,0,\,-1 \}$coincides with the virtual boson direction.

• Thrust T_c
  

  <B>B> (1)

• Thrust T_z
  

  <B>B> (2)

• Jet Broadening B_c
  

  <B>B> (3)

• Scaled Jet Mass $\rho_c$
  

  <B>B> (4)

A common characteristic of the mean event shape values $\langle 1 - T_c \rangle, \ \langle 1 - T_z \rangle, \langle \ B_c \rangle$ and $\langle \rho_c \rangle$ is the fact that they exhibit a clear decrease with rising Q, fig. 5. This is due to fact that the energy flow becomes more collimated along the event shape axis as Q increases, a phenomenon also observed in e⁺e^- annihilation experiments.

  

Figure 5: Mean event shape variables as a function of Q for a) $\langle 1 - T_c \rangle$, b) $\langle 1 - T_z \rangle / 2$, c) $\langle B_c \rangle$, and d) $\langle \rho_c \rangle$. H1 DIS $e\,p$ data ($\bullet$, errors include statistics and systematics) are compared with QCD fits (--) and second order QCD calculations ($\cdot\cdot\cdot$)

H1 showed by fitting to the data in fig. 5 all the event shape variables can be well described by just the first order power corrections $\propto 1/Q,$ without the need for any higher order corrections. The second order perturbative QCD parton predictions are also shown and their discrepancies with the data show that the power corrections are substantial at low values of Q, but become less important with increasing energy.

The analysis of the event shapes give results consistent with each other for $\bar{\alpha}_0,$ the power correction parameter thus supporting the prediction of universality [25], and also gives consistent values of $\alpha_s(M_Z).$ The results of the fit are $\bar{\alpha}_0 = 0.491 \pm 0.003~({\mbox{exp}}) \ ^{+0.079}_{-0.042}~({\mbox{theory}})$for the power correction parameter and $\alpha_s(M_Z) = 0.118 \pm 0.001~({\mbox{exp}}) \ ^{+0.007}_{-0.006}~({\mbox{theory}})$for the strong coupling constant in the $\overline{\mbox{MS}}$ scheme. These values are compatible with those extracted by e⁺e^- experiments [27]