A new cross term in the two-particle Hanbury-Brown-Twiss correlation function

Using two specific models and a model-independent formalism, we show that in addition to the usual quadratic “side”, “out” and “longitudinal” terms, a previously neglected “out-longitudinal” cross term arises naturally in the exponent of the two-particle correlator. Since its effects can be easily observed, such a term should be included in any experimental fits to correlation data. We also suggest a method of organizing correlation data using rapidity rather than longitudinal momentum differences since in the former every relevant quantity is longitudinally boost invariant. 25.70.Pq Typeset using REVTEX 1 The experimentally measured Hanbury-Brown Twiss (HBT) correlation between two identical particles emitted in a high energy collision defines a six dimensional function of the momenta p1 and p2 [1]. A popular way of presenting these is in terms “size parameters” derived from a gaussian fit to the data of the form [2–5] C(q,K) = 1± λ exp [ −q sR2 s(K)− q oR2 o(K)− q l R l (K) ] (1) where q = p1 − p2, K = 12(p1 + p2), the + (−) sign is for bosons (fermions), and the HBT cartesian coordinate system is defined as follows [6]: The “longitudinal” or ẑ (subscript l) direction is parallel to the beam; the “out” or x̂ (subscript o) direction is parallel to the component of K which is perpendicular to the beam; and the “side” or ŷ (subscript s) direction is the remaining direction. In this letter we assert that significantly more can be learned and better fits achieved if an “out-longitudinal” cross term is included in any gaussian fits to the data. In other words, we suggest that the data should be fit to a function with the following form C(q,K) = 1± λ exp [ −q sR2 s(K)− q oR2 o(K)− q l R l (K)− 2qoqlR ol(K) ] , (2) where R ol is a parameter which can be either positive or negative; we simply use the R 2 notation to denote the fact that it is has the dimension of an area. Since particles 1 and 2 are indistinguishable, the overall sign of q is irrelevant. The relative signs of the various components of q, however, are well defined physical quantities for any given pair. Our sign convention for q will be such that qs is always positive. We can thus unambiguously discuss correlations for negative as well as positive values of both qo and ql. To see how an “out-longitudinal” cross term arises in two-particle correlations, we use the following well established theoretical approximation [7,8] C(q,K) ≃ 1± | ∫ dxS(x,K) eiq·x|2 | ∫ d4xS(x,K)|2 (3) where q0 = E1 − E2, K0 = EK = √ m2 + |K|2. Here S(x,K) is a function which describes the phase space density of the emitting source. For pairs with |q| ≪ EK 2 q·x ≃ (βoqo + βlql) t− qox− qsy − qlz , (4) where βi = Ki/EK . As a simple example, we consider the following cylindrically symmetric gaussian emission function S(x,K) = f(K) exp [ − 2 + y 2R2 − z 2 2L2 − (t− t0) 2 2(δt)2 ] . (5) Using (3) and (4), it is easy to see that the corresponding correlation function takes the form: C(q,K) = 1± exp [ −q sR2 − q o(R2 + β o(δt))− q l (L + β l (δt))− 2qoqlβoβl(δt) ] . (6) For this model, the qoql cross term thus provides a measurement of the duration of particle emission (δt). One might think that this cross term is just a trivial kinematic effect which would not arise if the correlation were calculated in some more carefully chosen coordinate system or reference frame. For example, for spherically symmetric systems (L = R), it has been shown that the cross term vanishes if axes are chosen parallel and perpendicular to K (rather than parallel and perpendicular to the beam) [9]. The reader can verify, however, that for systems with L 6= R, the cross term does not vanish in these rotated coordinates, but rather measures the L − R asymmetry of the source. Another system that is often proposed is the Longitudinally Co-Moving System (LCMS), which is defined as the frame in which βl = 0 [10,4,5]. Glancing at (6), it naively appears that the cross term will vanish in this frame. Being more careful, however, one can see that after transforming to the LCMS frame (primed variables) t = γl(t− βlz) z = γl(z − βlt) , (7) where γl = 1/ √ 1− β l , tz terms arise in the transformed emission function S . These in turn lead to a nonvanishing cross term of the form