Log-periodic oscillations of transverse momentum distributions

Log-periodic oscillations of transverse momentum distributions arXiv:1403.3508v2 [hep-ph] 24 Mar 2014 Grzegorz Wilka , Zbigniew Włodarczykb a National Centre for Nuclear Research, Department of Fundamental Research, Hoża 69, 00-681 Warsaw, Poland b Institute of Physics, Jan Kochanowski University, Świȩtokrzyska 15, 25-406 Kielce, Poland Abstract Large pT transverse momentum distributions apparently exhibit power-like behavior. We argue that, under closer inspection, this behavior is in fact decorated with some log-periodic oscillations. Assuming that this is a genuine effect and not an experimental artefact, it suggests that either the exponent of the power-like behavior is in reality complex, or that there is a scale parameter which exhibits specific log-periodic oscillations. This problem is discussed using Tsallis distribution with scale parameter T . At this stage we consider both possibilities on equal footing. Keywords: scale invariance, log-periodic oscillation, p − p collisions Albeit both fits look pretty good, closer inspection shows that ratio of data/fit is not flat but shows some kind of clearly visible oscillations, cf. Fig. 2. It turns out that they cannot be eliminated by suitable changes of parameters (q, T ) or (m, T ) in Eqs. (1) or (2), respectively2. In what follows, we assume that this observation is not an experimental artifact but rather it represents some genuine 1  pT  1−q f (pT ) = C · 1 − (1 − q) (1) dynamical effect which is worth investigating in more deT tail. or, in the so called ”QCD inspired” Hagedorn form [9, 10] First notice, that to account for these fluctuations of (with parameters: m and T ): f (pT ) from Eq. (1) (or h (pT ) from Eq. (2)) the original Tsallis formula has to be multiplied by the following  1 pT −m ; m= h (pT ) = C 1 + . (2) factor (log-oscillating function): mT q−1   R(E) = a + b cos c ln(E + d) + f (3) For our purposes, these are equivalent (and we shall use them interchangeably.) They both represent the simplest way of describing the whole observed range of measured pT distributions. The best examples are the recent suc- approach. Nevertheless, it turns out that even from this kind of approach one can get a distribution with essentially only two parameters cessful fits [11] to very large pT data measured by the (not counting normalization) which closely resembles the usual Tsallis LHC experiments CMS [12], ATLAS [13] and ALICE distribution (1), albeit it is not identical with it [15]. Notice that at the midrapidity, i.e., [14] for pp collisions, see Fig. 1 1 . q for y ≃ 0, and for large transverse momenta, pT > M, For some time now it has been popular to fit the different kinds of transverse momentum spectra measured in multiparticle production processes to of a Tsallis formula [1] (cf., for example, [2, 3, 4, 5, 6, 7, 8]). It can be written in one of two recognized forms: either in original Tsallis one (with two parameters: q and T ), Email addresses: [email protected] (Grzegorz Wilk), [email protected] (Zbigniew Włodarczyk) 1 This is the usual domain reserved for the purely perturbative QCD Preprint submitted to Elsevier one has E = M 2 + p2T cosh(y) ≃ pT . 2 One has to realize that to really see these oscillations one needs rather large domain in pT . Therefore, albeit similar effects can be also seen at lower energies, they are not so pronounced as here and therefore will not be discussed at this point. September 25, 2021 2 1x10 1.6 1/2 s = 0.9 TeV 1/2 s = 7.0 TeV 0 1x10 1.5 -2 1.4 -4 1.3 -6 1.2 (2 pT) d N/dydpT 1x10 R = data/fit 1x10 -1 2 1x10 -8 1x10 -10 1x10 1.0 0.9 0.8 -14 0.7 1x10 1/2 s = 0.9 TeV 1.1 -12 1x10 (a) 0.6 -16 10 1 10 1 100 10 100 pT [GeV] pT [GeV] Figure 1: (Color online) Fit to large pT data for pp collisions at 0.9 and 7 TeV from CMS experiment using distribution (2) [12]. Parameters used are, respectively, (T = 0.135, m = 8) and (T = 0.145, m = 6.7). 1.6 1.5 (b) 1.4 1/2 s = 7 TeV 1.3 R = data/fit As recently shown in [16], such a factor, dressing the original power law distribution (in our case quasi-power law Tsallis distribution (1)), arises in a natural way if one allows the power index q to be complex3. For completeness, we shortly explain what this means. In general, if some function O(x) is scale invariant, i.e., if O(x) = µO(λx), then it must have a power law behavior, 1.2 1.1 1.0 0.9 0.8 0.7 0.6 O(x) = Cx−m with ln µ m= . ln λ 1 (4) Because one can write µλ−m = 1 = ei2πk , where k is an arbitrary integer, in general, m=− 2πk ln µ +i . ln λ ln λ 10 100 pT [GeV] Figure 2: (Color online) Fit to pT dependence of data/fit ratio for results from Fig. 1. Parameters of function R used (3) here are, respectively: a = 0.865, c = 2.1 for 0.9 TeV and a = 0.909, c = 1.86 for 7 TeV, whereas for both energies b = 0.166, d = 0.948 and f = −1.462. (5) As shown in [16], the evolution of the differential can write Eq. (1) in the form: d f (E)/dE of a Tsallis distribution f (E) with power index n performed for finite differences dE = α(nT + E) ln(1 − αn) 2π (where α < nT is another new parameter) results in the g(x) = x−mk , mk = − + ik . ln(1 + α) ln(1 + α) following scale invariant relation g[(1 + α)x] = (1 − αn)g(x) (6) (7) The power index in Eq. (7) (and in Eq. (1)) is therefore a complex number, the imaginary part of which signals where x = 1 + E/(nT ). This means that, in general, one a hierarchy of scales leading to the log-periodic oscillations. The meaning of the parameter α becomes clear by 3 There is vast literature of such situation in different branches of noticing that in the special case of k = 0, for which one rephysics, cf. [17] and other references [16]. covers the usual real power law solution, m0 corresponds 2 lows: m= 100 2π , d = nT, f = −c · ln(nT ). ln(1 + α) (10) Comparison of the fit parameters of the oscillating term R in Eq. (3) with Eq. (7) clearly shows that the observed frequency, here given by the parameter c, is more than an order of magnitude smaller than the expected value equal to 2π/ ln(1 + α) for any reasonable value of α. To explain this, notice that in our formalism leading to Eq. (9)) only one evolution step is assumed, whereas in reality we have a whole hierarchy of κ evolutions. This results (cf. [16]) in the scale parameter c being κ times smaller than in (9), a = w0 , b = w1 , c = 64 m=1/(q-1) 32 16 10 q from f( pT) [Wibig] f( pT) [CMS] f( mT) [NA49] m=-ln(1-n )/ln(1+ ) ; n=4 8 4 1 10 100 1000 1/2 s 10000 [GeV] Figure 3: (Color online) The energy dependence of m = m0 deduced from data [20, 12, 21]. c= 2π . κ ln(1 + α) (11) Experimental data indicate that κ ≃ 22 (for α ≃ 0.15 and to fully continuous scale invariance4. In this case one re- c ≃ 2 ). covers in the limit α → 0 the power n in the usual Tsallis distribution. However, in general one has 0.26 g(x) = X k=0 = 0.24
wk · Re x−mk = x−Re(mk ) X wk · cos [Im (mk ) ln(x)] . 0.22 (8) k=0 0.20 This is a general form of a Tsallis distribution for complex values of the nonextensivity parameter q. It consists of the usual Tsallis form (albeit with a modified power exponent) and a dressing factor which has the form of a sum of log-oscillating components, numbered by parameter k. Because we do not know a priori the details of dynamics of processes under consideration (i.e., we do not known the weights wk ), in what follows we use only k = 0 and k = 1 terms. We obtain approximately, 0.18 0.16 [Wibig] [CMS] [NA49] 1/2 3 =0.25-0.008ln[1+(s /200) ] 0.14 10 100 1000 1/2 s 10000 [GeV] Figure 4: (Color online) The energy dependence of parameter α present in m0 plotted in Fig. 3. From Eq.(7) we see that m0 > n. This suggests the following explanation of the difference seen between prediction from theory and the experimental data: the measurements in which log-periodic oscillations appear underestimate the true value that follows from the underlying dynamics which leads to the smooth Tsallis distribution. As an example consider the m0 dependence on α, assuming the initial slope n = 4 (this is the value of n expected from the pure QCD considerations for partonic interactions [15]). The energy behavior of the power index m0 ( " #)   E −m0 2π E  w0 + w1 cos . g(E) ≃ 1+ ln 1+ nT ln(1+α) nT (9) In this case one could expect that parameters in general modulating factor R in Eq. (3) could be identified as fol4 In this case power law exponent m0 still depends on α and   inn 4n2 + 3n − 1 α2 + creases with it roughly as m0 ≃ n + 2n (n + 1)α + 12   n 3 2 3 24 6n + 4n − n + 1 α + . . .. Notice also that α < 1/n. 3 we allow for an energy dependent noise, ξ(t, E)): in the Tsallis part is shown in Fig. 3, whereas the energy dependence of the parameter α contained in m0 is shown in Fig. 4. So far we attributed the observed log-periodic oscillations to the complex values of the power index m (i.e., to the complex nonextensivity parameter q)5 . However, this phenomenon can be also explained in a completely different way, namely by keeping the nonextensivity parameter q real (as in the original Tsallis distribution) but instead allowing the parameter T to oscillate in a specific way as displayed in Fig. 5. As seen there, the observed logperiodic oscillations of R can be reproduced by a suitable pT dependence of the scale parameter (the temperature) T , present in Tsallis distribution, here expressed by following a general formula (resembling Eq. (3), with generally energy dependent fit parameters (ā, b̄, c̄, d̄, f¯)): h   i T = ā + b̄ sin c̄ ln(E + d̄ + f¯ (12) dT 1 + T + ξ(t, E)T = Φ. dt τ For the time dependent E = E(t) it reads: dT dE 1 + T + ξ(t, E)T = Φ. dE dt τ dE E = + T. dt n E n T [GeV] +T  dT dE 1 + T + ξ(t, E)T = Φ. τ 0.150 0.145 and, after differentiating, to 0.140 0.130 1/2 s = 0.9 TeV 1/2 s = 7.0 TeV 10 (16) (17) #2 ! " d2 T 1 dT − ln E + Te e− ln E − + n d(ln E) d(ln E)2 # " dT 1 − ln E + − Te − − ξ(t, E) τ d(ln E) dξ(t, E) = 0. (18) +T d(ln E) 0.135 1 (15) Using now Eq. (15) one can write Eq. (14 as This can be subsequently transformed to ! 1 dT 1 + T e− ln E + T + ξ(t, E)T = Φ n d(ln E) τ 0.120 0.1 (14) In the scenario of preferential attachment (known from the growth of networks [19]) one expects that6 0.155 0.125 (13) 100 For large E (i.e., neglecting terms ∝ 1/E) one obtains the following equation for T : " # 1 1 d2 T dξ(t, E) dT + + ξ(t, E) +T = 0. (19) 2 n d(ln E) τ d(ln E) d ln E) pT [GeV] Figure 5: (Color online) The T = T (pT ) for Eq. (12) for which R = 1. Parameters used are: ā = 0.132, b̄ = 0.0035, c̄ = 2.2, d̄ = 2.0, f¯ = −0.5 for 0.9 TeV and ā = 0.143, b̄ = 0.0045, c̄ = 2.0, d̄ = 2.0, f¯ = −0.4 for 7 TeV. Now assume that noise ξ(t, E) increases logarithmically To explain such behavior, start with the well known with energy, [18] stochastic equation for the temperature evolution, ω2 which in the Langevin formulation has the form (in which ln E. (20) ξ(t, E) = ξ0 (t) + n 5 The possible dynamical implication of this fact, cf. for example [17, 16] and remarks in footnote 7, is outside of the scope of present paper. 6 Notice that in the usually used multiplicative noise scenario described by γ(t), not discussed here, one has dE dt = γ(t)E + ξ(t). 4 For this choice of noise Eq. (19) is just an equation for the damped hadronic oscillator and has a solution in the form of log-periodic oscillation of temperature with frequency ω: ( " # ) 1 ξ(t, E) T = C exp −n · + ln E · sin(ω ln E + φ). 2τ 2 (21) The phase shift parameter φ depends on the unknown initial conditions and is therefore an additional fitting parameter. Averaging the noise fluctuations over time t and taking into account that the noise term cannot on average change the temperature (cf. Eq. (13) in which hdT/dti = 0 for Φ = 0), i.e., that 1 + hξ(t, E)i = 0, (22) τ we have b′ T = ā + sin(ω ln E + φ). (23) n ′ The amplitude of oscillations, b /n, comes from the assumed behavior of the noise as given in Eq. (20). Notice that for large n, the energy dependence of the noise disappears (and because, in general, n decreases with energy, one can therefore expect only negligible oscillations for lower energies but increasing with the energy). This should now be compared with the parametrization of T (E) given by Eq. (12) and used to fit data in Fig.5. Looking at parameters we can see that only a small amount of T (of the order of b̄/ā ∼ 3%) comes from the stochastic process with energy dependent noise, whereas the main contribution emerges from the usual energy-independent Gaussian white noise. To conclude, the above oscillating T needed to fit the log-periodic oscillations seen in data can be obtained in yet another way. So far we were assuming that the noise ξ(t, E) has the form of Eq. (20) and, at the same time, we were keeping the relaxation time τ constant. However, it turns out that we could equivalently assume the energy E independent white noise, ξ(t, E) = ξ0 (t), but allow for the energy dependent relaxation time taken in the form of nτ0 . (24) τ = τ(E) = n + ω2 ln E This assumption corresponds to the following time evolution of the temperature, ! t −tω2 /n T (t) = hT i + [T (t = 0) − hT i]E exp − , (25) τ0 which is gradually approaching its equilibrium value hT i and reaches it more quickly for higher energies. To summarize, we have presented two possible mechanisms which could result in the log-periodic oscillations apparently present in data for transverse momentum distributions observed in LHC experiments. In both cases one uses a Tsallis formula (either in the form of Eq. (1) or Eq. (2)), with main parameters m - the scaling power exponent (or nonextensivity q = 1 + 1/m) and T - the scale parameter (temperature). In the first approach, our Tsallis distribution is decorated by the oscillating factor which emerges in a natural way in the case of complex power exponent m (or complex nonextensivity q)7 with the scale parameter T remaining untouched. In the second approach, it is the other way around, i.e., whereas m (or n as in Eq. (21)) remains untouched, the scale parameter T is now oscillating. From Eq. (23) one can see that T = T (n = 1/(1 − q), E) and as a function of nonextensivity parameter q it continues our previous efforts to introduce an effective temperature into the Tsallis distribution, T e f f = T (q), here in a much more general form as in [2] or [25]. The two possible mechanisms resulting in such T were outlined: the energy dependent noise connected with the constant relaxation time, or else the energy independent white noise, but with energy dependent relaxation time. We close by noting that, at the present level of investigation, we are not able to indicate which of the two possible mechanisms presented here (complex q or oscillating T ) and resulting in log-periodic oscillations is the preferred one. This would demand more detailed studies on the possible connections with dynamical pictures. For example, as discussed long time ago by studying apparently similar effects in some exclusive reactions using the QCD Coulomb phase shift idea [26]. The occurrence of some kind of complex power exponents was noticed there as well, albeit on completely different grounds than in our case. A possible link with our present analysis would be very interesting but would demand an involved and thor7 It is worth to mention at this point that complex q inevitably means also complex heat capacity C = 1/(1 − q) (c.f., [2, 22] and also [23]). Such complex (frequency dependent) heat capacities are widely known and investigated, see [24]. 5 ough analysis. [12] V. Khachatryan et al. (CMS Collaboration), JHEP 02 (2010) 041 and JHEP 08 (2011) 086; Phys. Rev. Lett. 105 (2010) 022002. Acknowledgments This research was supported in part by the National Science Center (NCN) under contract Nr [13] G. Aad et al. (ATLAS Collaboration), New J. Phys. 2013/08/M/ST2/00598. 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