Relativistic corrections of the fragmentation functions for a heavy quark to Bc and ${{B}_{\!\!{c}}^*}~$


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Deshan Yang and Wenjie Zhang. Relativistic corrections of the fragmentation functions for a heavy quark to Bc and ${{B}_{\!\!{c}}^*}~$[J]. Chinese Physics C, 2019, 43(8): 083101. doi: 10.1088/1674-1137/43/8/083101
Deshan Yang and Wenjie Zhang. Relativistic corrections of the fragmentation functions for a heavy quark to Bc and ${{B}_{\!\!{c}}^*}~$[J]. Chinese Physics C, 2019, 43(8): 083101.  doi: 10.1088/1674-1137/43/8/083101 shu
Received: 2019-04-19
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Relativistic corrections of the fragmentation functions for a heavy quark to Bc and ${{B}_{\!\!{c}}^*}~$

  • 1. School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
  • 2. Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

Abstract: In this paper, we compute the relativistic corrections of the fragmentation functions (FFs) for a heavy quark to Bc and $ B_c^* $ within the framework of non-relativistic QCD (NRQCD) factorization. The non-singlet and singlet DGLAP evolutions are also presented.


    1.   Introduction
    • One of the main fields of precision examination of the perturbative Quantum Chromodynamics (QCD) is the study of hadron production with momentum p ($ p^2 = m_H^2 $) at large transverse momentum $ p_T $ , where $ p_T^2 >> m_H^2 $. At the leading power of $ 1/p_T $, the fragmentation mechanism may be dominant so that the differential cross-section can be factorized as (for reviews, see [1-4])

      $\begin{split} {E_p}\frac{{{\rm d}{\sigma _{A + B \to H + X}}}}{{{{\rm d}^3}p}} \approx & \sum\limits_f {\int {\frac{{{\rm d}z}}{{{z^2}}}} } {D_{f \to H}}(z;\mu ){E_c}\\ & \times \frac{{{\rm d}{\sigma _{A + B \to f({p_c}) + X}}}}{{{{\rm d}^3}p}}\left( {{p_c} = \frac{1}{z}p} \right) ,\end{split}$


      where the parton-level differential cross-section $ {\rm d}\sigma_{A+B\to f(p_c)+X} $ contains the short-distance dynamics, and $ D_{f\to H}(z;\mu) $ is the so-called single parton fragmentation function (FF) of a hadron H from parton f , parametrizing the universal hadronization effects [5, 6]. Generally, the fragmentation functions are non-perturbative, but their renormalized scale-dependence is perturbatively calculable, and can be described by the famous Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations which also govern the renormalization group (RG) running of the parton distribution functions (PDFs)

      $\begin{split}{\mu ^2}\frac{\rm d}{{{\rm d}{\mu ^2}}}{D_{f \to H}}(z;\mu ) = &\sum\limits_{{f^\prime }} {\int_z^1 {\frac{{{\rm d}{z^\prime }}}{{{z^\prime }}}} } {P_{f \leftarrow {f^\prime }}}(z/{z^\prime },{\alpha _s}(\mu ))\\ &\times{D_{{f^\prime } \to H}}({z^\prime };\mu ) ,\end{split}$


      where $ P_{f\leftarrow f^\prime}(z,\alpha_s(\mu)) $ are the celebrated Altarelli-Parisi splitting functions [7, 8].

      For FFs of quarkonia, which are different from FFs of light mesons which rely completely on the dynamics in the non-perturbative regime of QCD, one believes that they can be further factorized into products of perturbatively calculable parts and non-perturbative behavior of the wave-functions of quarkonia at the origin, due to the nature of quarkonium as a non-relativistic bound state of a heavy quark and anti-quark. The standard theoretical tool to deal with the heavy quark bound state system is the NRQCD factorization [9], where all information about the hadronization of quarkonium is encoded in the NRQCD matrix elements. In the literature [10-16], various single-parton FFs of quarkonia have been calculated. Recently, such fragmentations for quarkonia have been calculated to a higher order of $ \alpha_s $ and v analytically or numerically [17-26] by implementing the state-of-art multi-loop calculation techniques.

      $ B_{c} $ mesons are unique as they consist of two heavy quarks of different flavor. Their production in high energy collisions is a hot topic in perturbative QCD since 1990s [27-42]. In these works, a lot of effort has been paid to FFs of $ B_c $ mesons, and most of the calculations were done in the framework of NRQCD factorization, similar to the corresponding calculations for quarkonia. It is worth noting that the first NLO calculation of FF for a heavy-quark to $ B_c^{(*)} $ meson, which is quite non-trivial, was carried out in [38].

      In this paper, we compute the relativistic corrections of FFs for a heavy-quark to $ B_c $ and $ B_c^* $ at the leading order of $ \alpha_s $. These corrections are equally important as the NLO QCD radiative corrections due to the NRQCD power counting rule. Similar results appeared in the Appendix of [15] as a by-product of the calculation of FFs for quarkonia. The differences between this paper and Ref. [15] are listed in the following: 1) we consider a different scheme for calculating the relativistic corrections for the S-wave heavy-quark bound system, in which the mass of $ B_c^{(*)} $ meson appears as an overall factor, which is different from the scheme adopted in [15] where the binding energy is treated as a part of non-relativistic corrections; 2) we consider the contribution from the color-octet NRQCD operators; 3) we calculate FF of a heavy quark to transversely polarized $ B_c^* $; 4) we further investigate the DGLAP evolution effects.

      The paper is organized in the following way. We first present in Sec. 2 the definition of FFs for b quark to $ B_c^{(*)} $ following Collins and Soper, and the desired NRQCD factorization formula. In Sec. 3, we illustrate the matching procedure, present the explicit expressions for the short-distance distributions, and compare them with those in literature. In Sec. 4, we present the non-singlet and singlet DGLAP evolutions of the obtained FFs. Finally, we summarize in Sec. 5.

    2.   Fragmentation function and its NRQCD factorization
    • In this section, we briefly review the definition of the fragmentation function which was given by Collins and Soper [5, 6], and present the NRQCD factorization formula for FFs of b-quark to $ B_c^{(*)} $.

      We adopt the following notation for the decompositions of momenta: the 4-velocity of $ B_c $ is $ v^{\mu} $ with $ v^2 = 1 $. We also use the same notation v for the non-relativistic expansion parameter, which is a typical value of the relative velocity of a quark and anti-quark inside a $ B_c $ meson. Depending on the context, one should not confuse these two. We also introduce two light-like vectors $ n^{\mu} $ and $ \bar{n}^{\mu} $ such that $ n^{2} = \bar{n}^{2} = 0 $ and $ \bar{n}\cdot n = 1 $. Any 4-vector $ a^{\mu} $ can be decomposed as $ a^{\mu} = a^{+}\bar{n}^{\mu}+a^{-}{n}^{\mu}+a_{\perp}^{\mu} $ ($ a^{+}\equiv n\cdot a $ and $a^{-}\equiv $$ \bar{n} \cdot a $) with $ n\cdot a_{\perp} = \bar{n}\cdot a_{\perp} = 0 $. Thus, $ a\cdot b = a^{+}b^{-}+a^{-}b^{+}+ $$a_{\perp}\cdot b_{\perp} $. For convenience, we set $ v^{\mu} = v^{+}\bar{n}^{\mu}+v^{-}n^{\mu} $. Obviously, $v^{+}v^{-} = $$ 1/2 $.

      The d-dimensional fragmentation function of b quark to $ B_c^{(*)} $ meson is defined as given by Collins and Soper in [5, 6],

      $\begin{split} {D_{b \to B_c^{(*)}}}(z;\mu ) = & \frac{{{z^{d - 3}}}}{{4{N_c}}}\sum\limits_X {\int\limits_{ - \infty }^\infty {\frac{{{\rm d}{x^ - }}}{{2\pi }}} } {{\rm e}^{ - {\rm i}{p^ + }{x^ - }/z}} \\ &\times{\rm{Tr}}\left[ {\not \!n\langle 0|W(0)b(0)|B_c^{(*)}(p),X\rangle}\right. \\ & \left. \times \langle B_c^{(*)}(p),X|\bar b({x^ - }n){W^\dagger }({x^ - })|0\rangle \right] \\ =& \frac{{{z^{d - 3}}}}{{4{N_c}}}\sum\limits_X \delta ({p^ + }/z - {p^ + } - p_X^ + )\\ &\times{\rm{Tr}}\left[ {\not\! n\langle 0|W(0)b(0)|B_c^{(*)}(p),X\rangle} \right. \\ & \left. \times\langle B_c^{(*)}(p),X|\bar b(0){W^\dagger }(0)|0\rangle \right] , \end{split}$


      where $ b(x) $ is the b quark field in QCD, and the Wilson line along a light-like path is defined as

      $W({x^ - }) \equiv {\rm{P}}\exp \left( { + {\rm i}{g_s}\int_{{x^ - }}^\infty {\rm d} s{A^ + }(sn)} \right),$


      in which $ g_s $ is the SU(3) gauge coupling and $ A_\mu (x)\equiv A^A_\mu(x) T^A $ ($ T^A $ are the generators of the SU(3) group in the fundamental representation). Since in this paper we never encounter the situation that would need the extra dimension to regulate the ultraviolet or infrared divergences, we always set $ d = 4 $.

      Since $ B_c^{(*)} $ can be regarded as a non-relativistic bound-state of b and anti-c quarks, the contribution of the heavy-quark mass may be further factorized. Therefore, up to $ {\cal O}(v^2) $, we get the factorization formula within the NRQCD factorization framework,

      $\begin{split} {D_{b \to {B_c}{(^{2s + 1}}{S_J})}}(z,\mu ) = &2{m_{{B_c}}}\Bigg\{ d_{2s + 1S_J^{[1]}}^{(0)}(z,\mu )\frac{{\langle {\cal O}_1^{{B_c}}{(^{2s + 1}}{S_J})\rangle }}{{{M^4}}} \\ &+ d_{2s + 1S_J^{[8]}}^{(0)}(z,\mu )\frac{{\langle {\cal O}_8^{{B_c}}{(^{2s + 1}}{S_J})\rangle }}{{{M^4}}}\\ & + d_{2s + 1S_J^{[1]}}^{(2)}(z,\mu )\frac{{\langle {\cal P}_1^{{B_c}}{(^{2s + 1}}{S_J})\rangle }}{{{M^6}}}\Bigg\} \\ &+ {\cal O}({v^4}) , \end{split}$


      where the mass scale $ M\equiv m_b+m_c $ is introduced to balance the mass dimensions so that the perturbatively calculated short-distance distributions $ d(z,\mu) $ are dimensionless, and the matrix elements of the singlet and octet NRQCD operators are defined as

      $\langle {\cal O}_1^{{B_c}}{(^1}{S_0})\rangle = \sum\limits_X {\langle 0|} \chi _c^\dagger {\psi _b}|{B_c} + X\rangle \langle {B_c} + X|\psi _b^\dagger {\chi _c}|0\rangle ,$


      $\langle {\cal O}_8^{{B_c}}{(^1}{S_0})\rangle = \sum\limits_X {\langle 0|} \chi _c^\dagger {T^A}{\psi _b}|{B_c} + X\rangle \langle {B_c} + X|\psi _b^\dagger {T^A}{\chi _c}|0\rangle ,$


      $\begin{split} \langle {\cal P}_1^{{B_c}}{(^1}{S_0})\rangle =& \frac{1}{2}\left[ {\sum\limits_X {\langle 0|} \chi _c^\dagger {\psi _b}|{B_c} + X\rangle} \right. \\ &\left.\times\langle {B_c} + X|\psi _b^\dagger {{\left( { - \frac{i}{2}\mathop {{ D}}\limits^{\leftrightarrow} } \right)}^2}{\chi _c}|0\rangle + {\rm{h}}.{\rm{c}}. \right],\end{split}$


      $\langle {\cal O}_1^{B_c^*}{(^3}{S_1})\rangle = \sum\limits_X {\langle 0|} \chi _c^\dagger {\sigma ^i}{\psi _b}|B_c^* + X\rangle \langle B_c^* + X|\psi _b^\dagger {\sigma ^i}{\chi _c}|0\rangle ,$


      $\begin{split} \langle {\cal O}_8^{B_c^*}{(^3}{S_1})\rangle =& \sum\limits_X {\langle 0|} \chi _c^\dagger {T^A}{\sigma ^i}{\psi _b}|{B_c} + X\rangle \\ &\times\langle {B_c} + X|\psi _b^\dagger {T^A}{\sigma ^i}{\chi _c}|0\rangle ,\end{split}$


      $\begin{split}\langle {\cal P}_1^{B_c^*}{(^3}{S_1})\rangle =& \frac{1}{2}\left[ {\sum\limits_X {\langle 0|} \chi _c^\dagger {\sigma ^i}{\psi _b}|B_c^* + X\rangle} \right. \\ & \left. \times\langle B_c^* + X|\psi _b^\dagger {\sigma ^i}{{\left( { - \frac{i}{2}\mathop {{D}}\limits^{\leftrightarrow} } \right)}^2}{\chi _c}|0\rangle + {\rm{h}}.{\rm{c}}. \right] ,\end{split}$


      where $ \chi_c $ and $ \psi_b $ are the two-component effective fields in NRQCD for $ \bar c $ and b quarks, respectively, $ \sigma^i $ is the i-th Pauli matrix, and $ {D} = {\nabla}-{ig_s}{A} $ is the covariant derivative. Note that all $ B_c^{(*)} $ states defined in (6-11) are non-relativistically normalized, while such states are relativistically normalized in the definition of FF in (3). The factor $ 2 m_{B_c} $ in the factorization formula (3) is just from the relativistic normalization condition of the single particle states.

      By using the vacuum saturation approximation, at the leading order of $ \alpha_s $, the matrix elements $ \langle {\cal O}_1\rangle $ can be related to the radial wave function of S-wave $ B_c $ meson at the origin $ R_{S}(0) $ in the color-singlet model through

      $\begin{split}&\langle {\cal O}_1^{{B_c}}{(^1}{S_0})\rangle \simeq \frac{{{N_c}}}{{2\pi }}|{R_S}(0){|^2} ,\\ &\langle {\cal O}_1^{{B_c}}{(^3}{S_1})\rangle \simeq (d - 1)\frac{{{N_c}}}{{2\pi }}|{R_S}(0){|^2} .\end{split}$


      However, there is no similarly simple relation for the matrix elements $ \langle {\cal O}_8\rangle $.

    3.   Computation of the fragmentation functions

      3.1.   Matching procedure

    • In the practical computation of the short-distance distributions $ d(z,\mu) $, we replace $ B_c^{(*)} $ with a color-singlet quark pair $ [b\bar c(^{2s+1}S_J^{[1]})] $,

      $|[b\bar c{(^{2s + 1}}S_J^{[1]})](p)\rangle = \sum\limits_{a,b = 1}^{{N_c}} {\frac{{{\delta _{ab}}}}{{\sqrt {{N_c}} }}} |{b^a}({p_b}),{\bar c^b}({p_c})\rangle ,$


      or a color-octet quark pair $ [b\bar c(^{2s+1}S_J^{[8]})] $

      $|[b\bar c{(^{2s + 1}}S_J^{[8]})](p)\rangle = \sum\limits_{a,b = 1}^{{N_c}} {\frac{{T_{ab}^A}}{{\sqrt {{T_F}} }}} |{b^a}({p_b}),{\bar c^b}({p_c})\rangle ,$


      where a, b are color indices, and $ T_F = 1/2 $ is the Dynkin index of the SU(3) group in the fundamental representation. We set the momenta as

      $\begin{split} &{p^\mu } = {M_H}{v^\mu } = p_b^\mu + p_c^\mu ,\;\;\;p_b^2 = m_b^2 ,\;\;p_c^2 = m_c^2 ,\\ & {p_b} = {m_b}{v^\mu } + {k^\mu } ,\;\;{p_c} = {m_c}{v^\mu } + {{\tilde k}^\mu } , \end{split}$


      where $ k^\mu $ and $ \tilde{k}^\mu $ are the residual momenta of b-quark and $ \bar c $-quark, respectively.

      At the leading order of $ \alpha_s $, we have

      $\begin{split} {D_{b \to [b\bar c]}}(z;\mu ) =& \frac{z}{{4{N_c}}}\sum\limits_{{\rm{colors\& spins}}} {\int {\frac{{{\rm d}{q^{\rm{ + }}}{{\rm d}^{\rm{2}}}{q_ \bot }}}{{{{({\rm{2}}\pi )}^{\rm{3}}}{\rm{2}}{q^ + }}}}} \\ &\times\theta ({q^{\rm{ + }}})\delta ({p^{\rm{ + }}}{\rm{/}}z{\rm{ - }}{p^{\rm{ + }}}{\rm{ - }}{q^{\rm{ + }}}){{\left| {\cal M} \right|}^{\rm{2}}} ,\end{split}$



      $\begin{split} {\left| {\cal M} \right|^2} = &{\rm{Tr}}\Big[ {\not \!\!n\langle 0|W(0)b(0)|[b\bar c](p),c(q)\rangle} \\ &\times\langle [b\bar c,c(q)](p),c(q)|\bar b(0){W^\dagger }(0)|0\rangle \Big] ,\end{split}$


      where $ q^2 = m_c^2 $. Schematically, this is a similar NRQCD factorization formula as in (5), when the $ B_c^{(*)} $ state is replaced by the corresponding $ b\bar c $ pair state.

      In the light-cone gauge,

      $\begin{split} &\langle {b^a}({p_b}),{{\bar c}^b}({p_c}),{c^c}(q)|{\left[ {\bar b(0){W^\dagger }(0)} \right]_d}|0\rangle = - g_s^2T_{ad}^AT_{cb}^A\\ &\left\{ {\frac{{{{\bar u}_c}(q){\gamma ^\mu }{v_c}({p_c}){{\bar u}_b}({p_b}){\gamma _\mu }(\not \!\!p \!+\! \not \!\!q \!+\! {m_b})}}{{[{{(p + q)}^2} - m_b^2][{{({p_c} + q)}^2}]}} - \frac{{{{\bar u}_c}(q)\not \!\!n{v_c}({p_c}){{\bar u}_b}({p_b})}}{{[{{({p_c} \!+\! q)}^2}][{{({p_c}\! +\! q)}^ + }]}}} \right\}, \end{split}$


      where we have used various equations of motion to simplify the spin structures. We also checked the expression of this matrix element in the covariant gauge, which is the same as the above expression in the light-cone gauge, as it should be.

      Since FF is defined in a Lorenz invariant way, we can do the calculation in any reference frame. For convenience, we choose the rest frame of the $ b\bar c $ pair, in which we have $ v^\mu = (1,\bf{0}) $ and

      ${k^\mu } = \left( {\frac{{{{{k}}^{{2}}}}}{{2{m_b}}},{{k}}} \right) ,\;\;\;{\tilde k^\mu } = \left( {\frac{{{{{k}}^{{2}}}}}{{2{m_c}}}, - {{k}}} \right) .$

      Generally, one applies the replacements

      $\begin{split}v({p_c})\bar u({p_b}) &\to {\Pi _0}({p_b},{p_c}) = \\ &i\frac{{\left( {{\rm{ }}{{\not \!\!p}_c} - {m_c}} \right){\gamma _5}(\not \!\!p + {E_b} + {E_c})\left( {{{\not \!\!p}_b} + {m_b}} \right)}}{{2\sqrt {2{E_b}{E_c}} \sqrt {({E_c} + {m_c})({E_b} + {m_b})} 2({E_b} + {E_c})}} ,\end{split}$


      $\begin{split}v({p_c})\bar u({p_b}) &\to {\Pi _1}({p_b},{p_c}) =\\ & - \frac{{\left( {{\rm{ }}{{\not \!\!p}_c} - {m_c}} \right){{\not \!\!\varepsilon }^*}({\rm{ }}\not \!\!p + {E_b} + {E_c})\left( {{\rm{ }}{{\not \!\!p}_b} + {m_b}} \right)}}{{2\sqrt {2{E_b}{E_c}} \sqrt {({E_c} + {m_c})({E_b} + {m_b})} 2({E_b} + {E_c})}} ,\end{split}$


      to project out the spin-singlet and spin-triplet parts, respectively. Here, $ \varepsilon^\mu $ denotes the polarization vector of the spin-triplet quark pair, and the energies are $ E_b = $$ \sqrt{m_b^2+{k}^2} $ and $ E_c = \sqrt{m_c^2+{k}^2} $. After expanding the resulting matrix element (18) to the second order in momentum k, one can extract the S-wave contributions by neglecting the first order terms in k (which contribute to the P-wave state only) and by making the replacement $ k^i k^j\to { k}^2 \delta^{ij}/3 $. This leads to the corresponding matrix elements of NRQCD operators at the leading order of $ \alpha_{s} $,

      $\begin{split}\langle {\cal O}_1^{b\bar c}{(^1}{S_0})\rangle \equiv &\sum\limits_X {\langle 0|} \chi _c^\dagger {\psi _b}|[b\bar c]{(^1}S_0^{[1]}) + X\rangle \\ &\times\langle [b\bar c]{(^1}S_0^{[1]}) + X|\psi _b^\dagger {\chi _c}|0\rangle = 2{N_c},\end{split}$


      $\begin{split}\langle {\cal O}_8^{b\bar c}{(^1}{S_0})\rangle \equiv &\sum\limits_X {\langle 0|} \chi _c^\dagger {\psi _b}|[b\bar c]{(^1}S_0^{[8]}) + X\rangle \\ &\times\langle [b\bar c]{(^1}S_0^{[8]}) + X|\psi _b^\dagger {\chi _c}|0\rangle = 2{N_c}{C_F},\end{split}$


      $\begin{split}\langle {\cal O}_8^{b\bar c}{(^1}{S_0})\rangle \equiv &\sum\limits_X {\langle 0|} \chi _c^\dagger {\psi _b}|[b\bar c]{(^1}S_0^{[8]}) + X\rangle \\ &\times\langle [b\bar c]{(^1}S_0^{[8]}) + X|\psi _b^\dagger {\chi _c}|0\rangle = 2{N_c}{C_F},\end{split}$


      $\begin{split}\langle {\cal O}_8^{b\bar c}{(^3}{S_1})\rangle \equiv &\sum\limits_X {\langle 0|} \chi _c^\dagger {\sigma ^i}{\psi _b}|[b\bar c]{(^3}S_1^{[8]}) + X\rangle \\ &\times\langle [b\bar c]{(^3}S_1^{[8]}) + X|\psi _b^\dagger {\sigma ^i}{\chi _c}|0\rangle \\ = &2(d - 1){N_c}{C_F},\end{split}$


      $\langle {\cal P}_1^{b\bar c}(n)\rangle = {{{k}}^2}\langle {\cal O}_1^{b\bar c}(n)\rangle ,\;\;n{ = ^1}S_0^{[1]} {,^3}S_1^{[1]} .$


      It is worth noting that we are calculating FF for $ B_{c}^{*} $ by using the replacement rules for the summation of polarization vectors

      $\sum\limits_{\lambda = 0, \pm 1} {{\varepsilon ^{*\mu }}} (p,\lambda ){\varepsilon ^\nu }(p,\lambda ) \to - {g^{\mu \nu }} + \frac{{{p^\mu }{p^\nu }}}{{{p^2}}} = - {g^{\mu \nu }} + {v^\mu }{v^\nu },$


      for the unpolarized case, and