arXiv:1711.09304v1 [math-ph] 25 Nov 2017
Relativistic Voigt profile for unstable particles in high energy physics Radoslaw A. Kycia1,2,a, Stanislaw Jadach3,b November 28, 2017 1 Cracow
University of Technology, Faculty of Physics, Mathematics and Computer Science, PL-31155, Krak´ow, Poland 2 The 3 The
Faculty of Science, Masaryk University, Kotlsk 2, 602 00 Brno, Czechia
Henryk Niewodnicza´ nski Institute of Nuclear Physics Polish Academy of Sciences ul. Radzikowskiego 152, 31-342 Krak´ow, Poland
a
[email protected], b
[email protected]
Abstract The Voigt profile is one of the most used special function in optical spectroscopy. In particle physics a version of this profile which originates from relativistic Breit-Wigner resonance distribution often appears, however, in the literature there is no strict derivation of its properties. The purpose of this paper is to define properly relativistic Voigt profile and describe it along the same way as their standard nonrelativistic counterparts. Key words: Voigt profile, Breit-Wigner distribution, relativistic Breit-Wigner distribution, the Dirac delta MSC 2010: 33E20
1
Introduction
In nonrelativistic particle physics the cross section of an unstable particles/resonances is described by the (nonrelativistic) Breit-Wigner distribution [7] σ0 (E; µ, Γ) :=
1 Γ 2π (E − µ)2 +
1
, Γ 2 2
(1)
where the real variable E > 0 is an energy of the resonance, the other real parameters are µ > 0 - the mass of unstable particle and Γ > 0 - the width of the resonance. This distribution in mathematical applications is called the Cauchy distribution, when normalized to unity. Relativistic counterpart to (1) is also called (relativistic) Breit-Wigner distribution and is of the form [7] σ2 (E; µ, Γ) :=
µΓ 1 . 2 2 π (E − µ )2 + (µΓ)2
(2)
In measurement of resonances obeying these distributions, the additional factor which has to be taken into account is a beam broadening effect or detector sensitivity function, which is usually of the form of the Gauss distribution (E−E0 )2 1 G(E − E0 ; σ) := √ e− 2σ2 , σ 2π
(3)
where two real parameters have the following meaning - E0 (positive real number) is the position of the accelerator beam energy maximum and σ > 0 - the dispersion of the beam or detector sensitivity around the maximum. The distribution which results from the original Breit-Wigner distribution and takes into account broadening effect of the Gaussian distribution has the form of a convolution integral Z ∞ σi (E ′ ; µ, Γ)G(E − E ′ ; σ)dE ′ , i = 0, 2. (4) Vi (E; µ, Γ, σ) := σi ∗ G = −∞
The resulting distribution for i = 0 (nonrelativistic case) is called the Voigt distribution (see section 7.19 of [6]) and its standard normal form is presented in the Appendix A. In the next part we define in the similar way, as we call it, the relativistic Voigt function which is (4) for i = 2 (relativistic case). This function is often used in High Energy Physics and, sadly to say, to our best knowledge was not extracted as a mathematical entity on its own, e.g., see excellent compendium of special functions [6]. This paper is intended to fill this gap. In the next section we define relativistic Voigt profile, outline its basic properties and define also duping function for the Voigt functions. In the Appendices technical details and derivations of these results are provided. Appendix A summarizes well-known properties of standard Voigt profile.
2
Relativistic Voigt profile
Relativistic Voigt distribution can be introduced in the similar manner as for classical Voigt profile summarized in in Appendix A. It is useful in considerations of relativistic unstable particles like the Higgs boson [4]. The equation (4) for i = 2 is of the form Z ∞ (E ′ −E)2 1 1 µΓ − 2σ 2 √ dE ′ . (5) e V2 (E; µ, Γ, σ) = ′2 2 2 2 −∞ π (E − µ ) + (µΓ) σ 2π
2
Introducing the variable
E − E′ t= √ , 2σ
(6)
we obtain V2 (E; µ, Γ, σ) =
µΓ √ πσ 2π
Z
2
∞
−∞
√
(E −
√ e−t √ σ 2dt. 2σt − µ)2 (E − 2σt − µ)2 + (µΓ)2 (7)
Defining new variables E−µ u1 := √ , 2σ
E+µ u2 := √ , 2σ
a :=
Γµ , 2σ 2
the distribution can be rewritten in the form Z 2 e−t 1 a ∞ dt. V2 (E; µ, Γ, σ) = 2 √ σ 2 π π −∞ (u1 − t)2 (u2 − t)2 + a2
(8)
(9)
New function which can be called relativistic line broadening function (it is an analogy to (23) presented below) can be defined as a H2 (a, u1 , u2 ) := π
Z
∞ −∞
2
e−t dt, (u1 − t)2 (u2 − t)2 + a2
(10)
and (9) can be rewritten in the following form V2 (E; µ, Γ, σ) =
H2 (a, u1 , u2 ) √ . σ22 π
(11)
The relativistic H2 depends on three parameters, not on two as nonrelativistic counterpart (23) Let us give some simple properties of H2 . Proposition 1 We have H2 (u1 , u2 , a) = H2 (u2 , u1 , a) H2 (−u1 , u2 , a) = H2 (u1 , −u2 , a) H2 (u1 , u2 , −a) = −H2 (u1 , u2 , a),
(12)
H2 (−u1 , −u2 , a) = H2 (u1 , u2 , a).
(13)
from which results that
These symmetry properties allows one to consider the function in the first octant {u1 > 0, u2 > 0, a > 0} and then propagate the values to other octants. From the Appendix B and C it results that
3
Proposition 2 For u1 6= u2 ± 2 2 (e−u1 + e−u2 ). |u1 − u2 |
lima→0± H2 (u1 , u2 , a) =
(14)
For u1 = u2 and nonzero a the H2 function is finite and when a → 0 it becomes unbounded. When |u1 | and |u2 | tends to infinity for a 6= 0 then as the integrand in (10) tends to zero, so H2 also vanishes. It is also true when a → 0± and |u1 − u2 | → ∞. Summing up, H2 for fixed u1 6= u2 is discontinuous and singular for u1 = u2 when a → 0. In Fig. 2 there is a plot of a few examples of the H2 function.
Figure 1: The relativistic Voigt distribution (10) for a few values of parameters. The values at a → 0, according to (43) and (44), are for H2 (a, 1, 0): 1+ 1e and for H2 (a, 1, −1): 1e . The function H2 (a, 1, 1) at a = 0 is unbounded. The next function that is important in application is the dumping function that shows how the maximum of (9) changes as a function of σ - the smearing of a particle beam or a detector sensitivity. In applications, this function allows one to estimate the maximum of the convolution when a given smearing is selected. We define V0 (µ; µ, Γ, σ) , (15) D0 (σ; Γ, µ) := σ0 (µ; µ, Γ) D2 (σ; Γ, µ) :=
V2 (µ; µ, Γ, σ) . σ2 (µ; µ, Γ)
(16)
These functions are normalized in such a way that for Γ 6= 0 and µ 6= 0 D0 (0; Γ, µ) = D2 (0; Γ, µ) = 1,
4
(17)
Figure 2: Dumping functions of (15) and (16).
Figure 3: Dumping functions (15) and (16) for a few values of µ parameter. It can be noted than when µ is increased then the departure of the relativistic D2 from corresponding nonrelativistic D0 is for higher values of σ. according to the (48) in Appendix B. Plots of these functions are presented in Fig. 2, 3 and 4. The relativistic line broadening function has an alternative representation Proposition 3 The relativistic line broadening function for a > 0 can be pre-
5
Figure 4: Dumping functions (15) and (16) for a few values of Γ parameter. As Γ increases the distribution D2 becomes more flat around σ = 0. sented as 1 H2 (a, u1 , u2 ) = π
Z
∞
−t2
dte
−∞
Z
∞
0
e−ax cos(x(t − u1 )(t − u2 ))dx
(18)
or alternatively H2 (a, u1 , u2 ) = π1 Re
R∞ 0
(u +u )2 x −ax+ 14 ix 4u1 u2 − 1 i+x2
dx √π√11−ix e
.
(19)
The proof will be given in Appendix D.
3
Conclusions
Definition of the relativistic Voigot profile and the line broadening function was provided along the same way as the original Voigt function. In addition, some properties and comparison of the dumping functions for both profiles was given. It is believed that this popular in use function will be included in modern tables of special functions.
Acknowledgments RK research was supported by grant MUNI/A/1103/2016 of Masaryk University.
6
A
Appendix - Voigt profile
In this section for the sake of completeness the definition of normalized form of the Voigt distribution of (4) for i = 0 will be provided, as a comparison and motivation for our studies. The results are standard and details can be found in, e.g., [8, 3, 10]. The normalized relativistic Voigot can be obtained in the following way. Starting from Z +∞ ′ 2 Γ 1 1 − (E−E2 ) 2σ √ V0 (E; µ, Γ, σ) = e dE ′ , (20) 2 σ 2π −∞ 2π (E ′ − µ)2 + Γ 2 upon introducing the new variables and constants E − E′ , t := √ 2σ
Γ a := √ , 2 2σ
we obtain 1 a V0 (E; µ, Γ, σ) = √ 2πσ π
Z
∞
−∞
E−µ u := √ , 2σ
(21)
2
e−t dt. (u − t)2 + a2
(22)
Defining new function(usually it is denoted by H(a, u)) a H0 (a, u) := π
Z
∞ −∞
2
e−t dt, (u − t)2 + a2
(23)
called line broadening function, the equation (22) can be written in the form H0 (a, u) V0 (E; µ, Γ, σ) = √ , 2πσ
(24)
which is similar to (11) when considering the relativistic case. Using analogous as in the Appendix B and rather standard considerations, namely the Proposition 6, one gets lima→0±
1 a = ±δ(t − u), π (t − u)2 + a2
(25)
where δ is the Dirac delta distribution, and as the result 2
lima→0± H0 (a, u) = ±e−u .
(26)
One can note that when comparing it with the relativistic version (14) there is no singularity here at a = 0 (as for u1 = u2 in the relativistic case). Line broadening functions are presented in Fig. 5. We can also provide alternative representation for (23) using the following Proposition from [8], which will be rewritten for relativistic Voigt profile in the Appendix D
7
Figure 5: Voigt distribution (23) for a few values of parameters. The limiting values at a → 0± is given by (26). Compare it with 2. Proposition 4 a = (t − u)2 + a2
Z
0
∞
e−ax cos((t − u)x)dx
(27)
for a > 0. The proof is as follows R ∞ −ax R∞ cos((t − u)x)dx = Re 0 e−ax ei((t−u) dx = 0 e −1 = Re −a+i(t−u) = (t−u)a2 +a2 .
(28)
This allows us to rewrite (23) in the following way
1 π
R∞ 2 R∞ H0 (a, u) = π1 −∞ dte−t 0 dxe−ax cos((t − u)x) = R∞ R −ax ∞ dte−t2 (cos(ux) cos(ut) + sin(ux) sin(xt)) = 0 dxe −∞ R∞ R∞ 2 = π1 0 dxe−ax cos(ux) −∞ dte−t cos(xt) = R∞ x2 √1 dxe−ax− 4 cos(ux). π 0
Switching to complex representation we finally obtain Z ∞ x2 1 H0 (a, u) = Re √ dxe−ax+iux− 4 . π 0 This can be converted to the Faddeeva function described in [6].
8
(29)
(30)
B
Appendix - Limiting value for a → 0
In this section the normalization of (10) in the limit a → 0 will be derived. Define Z dt a ∞ . (31) I2 (u1 , u2 , a) := 2 π −∞ (t − u1 ) (t − u2 )2 + a2 As an introduction consider the following Proposition 5 For u1 6= u2 we have lima→0+ I2 (u1 , u2 , a) = lima→0− I2 (u1 , u2 , a) =
2 |u1 −u2 | −2 |u1 −u2 |
(32)
The proof is simple application or the Cauchy residue theorem [1]. The zeros p of the integrand denominator are located at 12 (u1 + u2 ± ±4ia + (u1 − u2 )2 ). Assume that a > 0 and consider the contour of integration in the complex plane presented in Fig. B. We have
Figure 6: Countour of integration in the complex plane for (31. The dots show the poles and R → ∞. RR
a dt −R π (t−u1 )2 (t−u2 )2 +a2
2πi
P
+
R
dt a a π (t−u1 )2 (t−u2 )2 +a2
= (33)
a 1 i resi π (t−u1 )2 (t−u2 )2 +a2 ,
where the second integral is over the half circle of the radius R in the upper complex half-plane and the sum is over the residues p p inside the contour, i.e., at 1 2 ) and at 1 (u + u − − u ) 4ia + (u −4ia + (u1 − u2 )2 ). Noting (u + u + 1 2 1 2 1 2 2 2 that in the limit R → ∞ the integral over the semicircle vanishes one has from the Cauchy residue theorem that 1 −4ia+(u1 −u2 )2
I2 (u1 , u2 , a) = √
9
+√
1 . 4ia+(u1 −u2 )2
(34)
Taking the limit a → 0+ one gets the first result of (32). The same result can be obtained if the circle in the lower complex semiplane is used. In order to prove the second statement it should be assumed that a < 0. Then the poles in the contour are interchanged with those that are outside the contour and as a result it generates additional minus sign. The second equation of (32) can be also deduced from the first one and the fact that I(u1 , u2 , −a) = −I(u1 , u2 , a). This ends the proof. It is worth noting that from the above reasoning using the same contour of integration we have Corollary 1 For an analytic function with no poles in the upper complex halfplane φ(t) such that lim|t|→∞ |φ(t)| = 0 we have Z ∞ a φ(t) ±1 lim dt = (φ(u1 ) + φ(u2 )), (35) 2 2 2 ± |u1 − u2 | a→0 −∞ π (t − u1 ) (t − u2 ) + a when u1 6= u2 . The Corollary cannot be used for φ(t) being the Gaussian function (3) as it is unbounded for Im(t) → ±∞, however, the behaviour from the Corollary is also valid for the Gaussian as it will be shown below. The obstacle is as follows, if one would like to use in the proof the technique of the Cauchy residue theorem it would require some complicated contour of integration encloses the poles of (31) π 3πthat and does not contain the sector of Arg(t) ∈ 4 , 4 . In order to overcome these difficulties a different approach will be presented that uses strengthen classical theorem from the Theory of Distributions [9]. This theorem uses the Gaussian function that defines series convergent to the Dirac delta distribution1 . It uses the idea of the class of functions that converges in distributional sense to the Dirac delta distribution, see e.g., §1.17 of [6] or [5, 2, 11]. Such series converges uniformly to zero except at one point, which occurs to be the zero of the argument of the Dirac delta distribution. The case of (10) is similar to the Dirac delta series in the limit a → 0± , except of the fact that the there are two singular points at u1 and u2 , and the function is discontinuous at a = 0 . We prove the following Proposition, which is significant generalization of those from [2, 11] (see Theorem 3.1.1 in [2] or Theorem 2.3.2 of [11]). Proposition 6 Let {fν } will be a sequence of locally integrable functions on R such that P.1 There is a finite set of points {xi }N i=1 and fixed sets of positive numbers that all the integrals {Xi }N i=1 Z |fµ (x)|dx, µ = 1, 2, . . . (36) |x−xi |:= −∞ δ(x)φ(x)dx := φ(0), i.e., it selects the value of φ at x = 0.
10
are uniformly bounded by a positive constant K for all ν. P.2 The sequence {fν } is uniformly convergent to zero on every finite interval not containing the points {xi }N i=1 . P.3 For all finite {τi }N i=1 such that the sets |x−xi | < τi have no points in common the integrals convergent to {αi }N i=1 Z fν dx → αi (37) |x−xi |:= R fν (x)φ(x)dx = N i=1 |x−xi | 0, i.e., −a−i(t−u1 )(t−u2 ) 1 = −1 a Re (t−u1 )2 (t−u2 )2 +a2 = (t−u1 )2 (t−u2 )2 +a2 −1 1 a Re −a+i(t−u1 )(t−u2 )
1 a
=
R ∞ x(−a+i(t−u )(t−u )) 1 1 2 dt a Re 0 e
R∞ 0
(53) =
e−ax cos(x(t − u1 )(t − u2 ))dt.
Introducing this into H2 we obtain double integral representation Z ∞ Z 1 ∞ −t2 H2 (a, u1 , u2 ) = e−ax cos(x(t − u1 )(t − u2 ))dx dte π −∞ 0
13
(54)
or alternatively H2 (a, u1 , u2 ) = π1 Re
R∞ 0
dte−ax
R∞
−t2 +ix(t−u1 )(t−u2 )) dx −∞ e
(u +u )2 x R∞ −ax+ 14 ix 4u1 u2 − 1 i+x2 1 1 √ √ . π Re 0 dx π 1−ix e
= (55)
These expressions is, however, more complicated comparing to the original one for H2 , i.e., equation (10).
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