Next-to-leading power corrections to event-shape variables

Abstract

We investigate the origin of next-to-leading power (NLP) corrections to the event shapes’ thrust and c-parameter, at next-to-leading order. For both event shapes, we trace the origin of such terms in the exact calculation, and compare with a recent approach involving the eikonal approximation and momentum shifts that follow from the Low–Burnett–Kroll–Del Duca (LBKD) theorem. We assess the differences both analytically and numerically. For the c-parameter both exact and approximate results are expressed in terms of elliptic integrals, but near the elastic limit it exhibits patterns similar to the thrust results.

Appendices

Appendix A. Transformation of incomplete elliptic integrals

In this Appendix, we demonstrate how we handle the incomplete elliptic integrals that appear in the expression of c-parameter distribution. The final expression for the c-parameter distribution is written in a compact manner in eq. (83), where K, E and $\Pi$ are the complete elliptic integrals of the first, second and third kinds, respectively given in eq. (84). However, when we perform integration over final variable y in eqs (78), (97) and (108), this integration produces incomplete elliptic integrals F, E and $\Pi$. These incomplete elliptic integrals are later converted into complete integrals to arrive at eq. (83), as first written in [110]. The three kinds of incomplete elliptic integrals appearing in c-parameter distribution are

$$\begin{aligned}&F[\phi , m]= \int _{0}^{\phi } \textrm{d} \theta \frac{1}{\sqrt{1-m \sin ^2\theta }}\, \nonumber \\&\qquad \qquad \,\,= \int _{0}^{\sin \phi } \frac{\textrm{d}t}{\sqrt{(1-t^2)(1-mt^2)}} \,, \end{aligned}$$

(A.1)

$$\begin{aligned}&E[\phi , m]= \int _{0}^{\phi } \textrm{d} \theta \sqrt{1-m \sin ^2\theta }\, \nonumber \\&\qquad \qquad \,\,=\int _{0}^{\sin \phi } \textrm{d}t \sqrt{\frac{1-mt^2}{1-t^2}} \,, \end{aligned}$$

(A.2)

$$\begin{aligned}&\Pi [n,\phi , m]= \int _{0}^{\phi } \textrm{d} \theta \frac{1}{(1-m\sin ^2\theta )\sqrt{1-m\sin ^2\theta }} \, \nonumber \\&\qquad \qquad \qquad \!=\int _{0}^{\sin \phi } \frac{\textrm{d}t}{(1-nt^2)\sqrt{(1-t^2)(1-mt^2)}} \,. \end{aligned}$$

(A.3)

Here, $\phi , m$ and n are called the amplitude, parameter and characteristic of the elliptic integrals, respectively. Equation (84) gives their respective complete forms. The corresponding transformation into a complete elliptic integral can be performed using the rules

$$\begin{aligned}&F[\phi ,m]=K[m]\,, \nonumber \\&E[\phi ,m]=E[m]\,, \\&\Pi [n,\phi , m]=\Pi [n, m]\,.\nonumber \end{aligned}$$

(A.4)

The above transformation is only possible when the amplitude $\phi =\pi /2$. The indefinite integration of eqs (78), (97) and (108) results in multiple incomplete elliptic integrals with only two unique amplitudes in their arguments, namely $\phi _1(c,y)$ and $\phi _2(c,y)$, given by

$$\begin{aligned} \phi _1(c,y)&=\left( \frac{-1+\sqrt{1-8c}-4c+8c/y}{2\sqrt{1-8c}}\right) ^{1/2}\,, \end{aligned}$$

(A.5)

$$\begin{aligned} \phi _2(c,y)&=\left( \frac{1+\sqrt{1-8c}+4c-8c/y}{2\sqrt{1-8c}}\right) ^{1/2}\,. \end{aligned}$$

(A.6)

The amplitudes $\phi _1(c,y)$ and $\phi _2(c,y)$ are present in the arguments of all three kinds of incomplete elliptic integrals. If we directly substitute the upper limit ($y_2$) after integration, then none of the incomplete elliptic integrals with the amplitude $\phi _1(c,y)$ can be reduced to a complete elliptic integral since

$$\begin{aligned} \phi _1(c,y) \vert _{y =y_2}=0\,. \end{aligned}$$

(A.7)

The transformation to complete elliptic integrals holds only when $\phi _1 =\pi /2$ as given in eq. (A.4). However, the incomplete elliptic integrals with the amplitude $\phi _2(c,y),$ upon substitution of the upper limit, can be directly reduced to a complete elliptic integral as

$$\begin{aligned} \phi _2(c,y) \vert _{y \,=\,y_2}=\frac{\pi }{2}\,. \end{aligned}$$

(A.8)

We observe a similar but opposite behaviour when we substitute the lower limit ($y_1$) into the integration result. This time, the incomplete elliptic integrals with the amplitude $\phi _1(c,y)$ can be reduced to the complete elliptic integral as

$$\begin{aligned} \phi _1(c,y) \vert _{y \,=\,y_1}=\frac{\pi }{2}\,, \end{aligned}$$

(A.9)

while the elliptic integrals with amplitude $\phi _2(c,y)$ cannot be reduced to the complete elliptic integral as

$$\begin{aligned} \phi _2(c,y) \vert _{y \,=\,y_1}=0\,. \end{aligned}$$

(A.10)

In order to resolve the issue of transforming the incomplete elliptic integrals to complete elliptic integrals, we modify the upper and lower limits of y integration as

$$\begin{aligned} (y_1, y_2) \rightarrow (y_1+e, y_2+e)\, , \end{aligned}$$

(A.11)

where e is an infinitesimal real off-set parameter that will be taken to zero at the end. The relative sign of e does not affect the final expression for the c-parameter distribution, which is independent of this off-set parameter. When we substitute the upper limit of integration $y_2$ with the off-set parameter as defined in the above expression, the amplitudes $\phi _1(c,y)$ and $\phi _2(c,y)$ modify and have the following form:

$$\begin{aligned}&\phi _1(c,y,e) \vert _{y \,=\,y_2+e}\nonumber \\&\, = \left( \frac{\left( -4 c+\sqrt{1-8 c}-1\right) (c+1) e}{\sqrt{1-8 c} \left( 2 c (e+2)+\sqrt{1-8 c}+2 e+1\right) }\right) ^{1/2} \,, \end{aligned}$$

(A.12)

$$\begin{aligned}&\phi _2(c,y,e) \vert _{y \,=\,y_2+e}\nonumber \\&\, =\left( \frac{-\frac{16 (c\!+\!1) c}{2 c (e\!+\!2)\!+\!\sqrt{1-8 c}\!+\!2 e\!+\!1}+4 c+\sqrt{1-8 c}+1}{2 \sqrt{1-8 c}}\right) ^{1/2}\,. \end{aligned}$$

(A.13)

Similarly, from the lower limit, the amplitudes take the form

$$\begin{aligned}&\phi _1(c,y,e) \vert _{y \,=\,y_1+e}\nonumber \\&\quad = \left( \frac{\frac{16 (c+1) c}{2 c (e+2)-\sqrt{1-8 c}+2 e+1}\!-\!4 c\!+\!\sqrt{1-8 c}\!-\!1}{2 \sqrt{1-8 c}}\right) ^{1/2} \,, \end{aligned}$$

(A.14)

$$\begin{aligned}&\phi _2(c,y,e) \vert _{y \,=\,y_1+e}\nonumber \\&\quad =\left( -\frac{(c+1) \left( 4 c+\sqrt{1-8 c}+1\right) e}{\sqrt{1\!-\!8 c} \left( -\!2 c (e\!+\!2)\!+\!\sqrt{1\!-\!8 c}\!-\!2 e\!-\!1\right) }\right) ^{1/2}\,. \end{aligned}$$

(A.15)

Note that, it is necessary to use this parameter because the straightforward substitution of limits did not allow the transformation of every incomplete elliptic integral. Let us consider a few examples to demonstrate how this off-set parameter solves the problem with incomplete elliptic integrals that cannot be reduced into complete elliptic integrals as their amplitude $\phi \ne \pi /2$. We categorise all the elliptic integrals appearing after y integration into two classes according to their amplitudes $\phi$ as (i) non-reducible incomplete elliptic integrals $(\phi \ne \pi /2)$ and (ii) reducible incomplete elliptic integrals $(\phi = \pi /2)$.

1.1 Appendix A.1 Non-reducible incomplete elliptic integrals

Here we consider an incomplete elliptic integral that appears from the upper limit contributions, and the expression of the elliptic integral is

$$\begin{aligned}&E[\phi _1(c,e),m_1(c)]\nonumber \\&=\!E\!\left[ \!\sin ^{-1}\!\left( \frac{\left( -\!4 c\!+\!\sqrt{1\!-\!8 c}\!-\!1\right) \! (c\!+\!1) e}{\sqrt{1\!-\!8 c} \!\left( 2 c (e\!+\!2)\!+\!\sqrt{1\!-\!8 c}\!+\!2 e\!+\!1\right) }\!\right) ^{1/2},\right. \nonumber \\&\quad \left. \frac{2\sqrt{1-8c}}{1+\sqrt{1-8c}-4c-8c^2} \right] \, , \end{aligned}$$

(A.16)

where $m_1$ is given in eq. (85). The amplitude of this elliptic function is

$$\begin{aligned}&\phi _1(c,e) \nonumber \\&\quad = \sin ^{-1}\left( \frac{\left( -4 c\!+\!\sqrt{1\!-\!8 c}\!-\!1\right) (c\!+\!1) e}{\sqrt{1\!-\!8 c} \left( 2 c (e\!+\!2)\!+\!\sqrt{1\!-\!8 c}\!+\!2 e\!+\!1\right) }\right) ^{1/2}\,. \end{aligned}$$

(A.17)

The elliptic integral in eq. (A.16) cannot be reduced to a complete elliptic integral as in the limit $e \rightarrow 0$ the amplitude $\phi _1(c,e) \rightarrow 0$. However, we can expand this elliptic integral in powers of e around $e = 0$, and upon expanding we get

$$\begin{aligned} E[\phi _1(c,e),m_1(c)]&=\sqrt{\frac{\left( -4 c+\sqrt{1-8 c}-1\right) (c+1)}{\sqrt{1-8 c} \left( 4 c+\sqrt{1-8 c}+1\right) }}\nonumber \\&\quad \times \sqrt{e} +\mathcal {O}(e^{3/2})\,. \end{aligned}$$

(A.18)

The expansion’s first term, proportional to $e^{1/2}$, is the only significant term in the limit when $e\rightarrow 0$.

The next term in the above expression is proportional to $e^{3/2}$, and can be dropped. The small-e expression for these specific incomplete elliptic integrals occurring is then stored in the form

$$\begin{aligned}&E[\phi _1(c,e),m_1(c)] \nonumber \\&\quad =\,\sqrt{\frac{\left( -4 c+\sqrt{1-8 c}-1\right) (c+1)}{\sqrt{1-8 c} \left( 4 c+\sqrt{1-8 c}+1\right) }} \sqrt{e}\,. \end{aligned}$$

(A.19)

Now, we shift our attention toward the elliptic integrals that appears from the lower limit contribution, one such incomplete elliptic integral is

$$\begin{aligned}&F[\phi _2(c,e),m_1(c)] \nonumber \\&\quad \!\!\!=\!F\!\left[ \!\sin ^{-1}\!\!\left( \!\frac{(c+1) \left( 4 c+\sqrt{1-8 c}+1\right) e}{\sqrt{1\!-\!8 c} \left( 2 c (e\!-\!2)\!+\!\sqrt{1\!-\!8 c}\!+\!2 e\!-\!1\!\right) }\!\right) ^{\!1/2}\!\!, \right. \nonumber \\&\left. \qquad \frac{2\sqrt{1-8c}}{1+\sqrt{1-8c}-4c-8c^2} \right] \,. \end{aligned}$$

(A.20)

The above elliptic integral cannot be reduced into a complete elliptic integral as the amplitude $\phi _2(c,e)$ is

$$\begin{aligned}&\phi _2(c,e) \nonumber \\&\quad =\, \sin ^{-1}\!\left( \frac{(c\!+\!1) \left( 4 c\!+\!\sqrt{1\!-\!8 c}\!+\!1\!\right) e}{\sqrt{1\!-\!8 c} \!\left( 2 c (e\!-\!2)\!+\!\sqrt{1\!-\!8 c}\!+\!2 e\!-\!1\right) }\right) ^{1/2}\,. \end{aligned}$$

(A.21)

and in the limit $e \rightarrow 0$ the amplitude $\phi _2(c,e) \rightarrow 0$. Following the similar procedure as in eq. (A.19), the elliptic integral in eq. (A.20) is expanded around $e = 0$, to get

$$\begin{aligned}&F[\phi _2(c,e),m_1(c)]\nonumber \\&\quad =\!\sqrt{\frac{(c+1) \left( 4 c+\sqrt{1-8 c}+1\right) }{\sqrt{1\!-\!8 c} \!\left( -\!4 c\!+\!\sqrt{1\!-\!8 c}\!-\!1\right) }} \sqrt{e}\, . \end{aligned}$$

(A.22)

Fig. 8

In (a) and (b), the thrust and c-parameter distributions, respectively are plotted. We show the exact result (solid red curves) given in eqs (28) and (83), the shifted approximation result (dashed blue curves) up to NLP terms given in eqs (47) and (101) for $\tau$ and c respectively. We also plot the eikonal distributions (dotted purple curves) given in eqs (A.27) and (A.28).

The coefficients of these elliptic functions depend on e when the integration limits are modified in accordance with eq. (A.11). When we expand our final result in powers of e, the negative powers of $\sqrt{e}$ from the coefficients combine with the positive powers of $\sqrt{e}$ from the stored expressions of non-reducible incomplete elliptic integrals to yield a few terms independent of e. Subsequently, e can be set to zero.

1.2 Appendix A.2 Reducible incomplete elliptic integrals

The category of incomplete elliptic integrals, which can be reduced to complete elliptic integrals is easier to handle as compared with non-reducible ones. An elliptic integral appearing from the upper limit contribution is

$$\begin{aligned}&E[\phi _2(c,e),m_1(c)] \nonumber \\&\,=\!E\!\left[ \!\sin ^{-1}\!\left( \!\frac{-\!\frac{16 (c\!+\!1) c}{2 c (e\!+\!2)\!+\!\sqrt{1\!-\!8 c}\!+\!2 e\!+\!1}\!+\!4 c\!+\!\sqrt{1\!-\!8 c}\!+\!1}{2 \sqrt{1\!-\!8 c}}\!\right) ^{\!\!\!1/2}, \right. \nonumber \\&\qquad \!\left. \frac{2\sqrt{1-8c}}{1+\sqrt{1-8c}-4c-8c^2} \right] \, , \end{aligned}$$

(A.23)

where the amplitude $\phi _2(c,e)$ reads as

$$\begin{aligned}&\phi _2(c,e) \nonumber \\&\,= \!\sin ^{-1}\!\!\left( \!\frac{\!-\frac{16 (c\!+\!1) c}{2 c (e\!+\!2)\!+\!\sqrt{1\!-\!8 c}\!+\!2 e\!\!+\!1}\!+\!4 c\!+\!\sqrt{1\!-\!8 c}\!+\!1}{2 \sqrt{1\!-\!8 c}}\right) ^{\!\!1/2}\,. \end{aligned}$$

(A.24)

In the limit $e \rightarrow 0$ the amplitude $\phi _2(c,e) \rightarrow \pi /2$ and this incomplete elliptic integral can be reduced directly to a complete elliptic integral using the reduction formula in eq. (A.4)

$$\begin{aligned} E[\phi _2(c,e),m_1(c)] \,=\,E[m_1(c)] \, . \end{aligned}$$

(A.25)

In this way, all reducible incomplete elliptic integrals are substituted with their corresponding complete elliptic integrals. Upon substituting all the incomplete elliptic integrals, our final result is expanded in powers of e. No negative powers occur and e can be taken to zero, yielding eq. (83).

Appendix B. Eikonal approximation and result summary

1.1 Appendix B.1 Eikonal approximation to the thrust and c-parameter distribution

In §3.3 and §4.3, we compared the shifted approximation with the LP expression of the exact distribution in figures 3a and 7a, for the thrust and c-parameter distribution, respectively. Here we add the results from the simpler eikonal approximation for comparison. The eikonal case follows from the approximated matrix element squared and is given by the first term in eq. (48) as

$$\begin{aligned}&\overline{\sum } |\mathcal {M}_{\text {eik}}(x_1,x_2)|^2 \,= \, 8(e^2e_q)^2 g_s^2 C_F N_c \frac{1}{3Q^2} \nonumber \\&\quad \times \left( \frac{2}{(1-x_1)(1-x_2)}\right) \,. \end{aligned}$$

(A.26)

Using the above expression and the exact definitions of thrust and c-parameter in eqs (10) and (69), one finds for their respective distributions

$$\begin{aligned}&\frac{1}{\sigma _{0}(s)} \frac{\textrm{d}\sigma }{\textrm{d}\tau } \Bigg \vert _{\text {NLO}} \!=\!\frac{2 \alpha _{s}}{3 \pi }\left( \frac{-4\log {\tau }}{\tau }\!-\!8\!-\!4\log {\tau } \!+\! \mathcal {O}(\tau ) \! \right) , \end{aligned}$$

(A.27)

$$\begin{aligned}&\frac{1}{\sigma _{0}(s)} \frac{\textrm{d}\sigma }{\textrm{d}c} \Bigg \vert _{\text {NLO}} \!=\!\frac{2 \alpha _{s}}{3 \pi }\left( \frac{-4\log {c}}{c}\!-\!12\!-\!8\log {c} \!+\! \mathcal {O}(c) \! \right) \,. \end{aligned}$$

(A.28)

The eikonal matrix element squared generates LL terms at LP correctly, together with some LL and NLL at NLP. It does not capture any NLL term at LP since it lacks contributions from hard-collinear gluon emission. In figures 8a and 8b, we plot these eikonal results together with the thrust and c-parameter distributions computed from the exact approach as given in eqs (28) and (83) along with the shifted approximation results up to NLP from eqs (47) and (101). Clearly for both event shapes, the shifted kinematics methods provides a significantly better approximation than the eikonal approximation.

Table 1 Result of the thrust and the c-parameter distributions calculated using the exact definition of event-shape variables and four different definitions of matrix elements.

1.2 Appendix B.2 Table of results

In table 1, we summarise our results for the thrust and the c-parameter using different approximations of the matrix elements squared: eikonal, exact, shifted kinematics, as well as the remainder/soft quark part.