Froissaron and Maximal Odderon with spin-flip in pp and $\bar{p}p$ high energy elastic scattering

Abstract

The scattering amplitude with spin-non-fiip and spin-flip components represented by Froissaron, Maximal Odderon as well as by standard Regge poles contributions is considered. From the fit to the data of pp and $\bar{p}p$ scattering at high energy and not too large momentum transfers we found that this model taking into account the spin is available to describe not only the differential, total cross section and $\rho$, but also the existing experimental data on polarization. It allows to make some predictions about spin effects at high energies.

Appendices

Appendix A: Simplified FMO model

The simple and dipole Pomerons (Odderons) for both components have a conventional form:

$$\begin{aligned}&P^{(S)}_{1}(s,t)=-g^{(S)}_{1,p}{\tilde{s}}^{\alpha _p(t)}e^{\beta ^{(S)}_{1,p}\tau _p}, \nonumber \\&P^{(D)}_{1}(s,t)=-g^{(D)}_{1,p} \xi {\tilde{s}}, ^{\alpha _p(t)}e^{\beta ^{(D)}_{1,p}\tau _p},\nonumber \\&\tau _p=2m_\pi ^2-\sqrt{4m_\pi ^2-t}, \end{aligned}$$

(A.1)

$$\begin{aligned} O^{(S)}_{1}(s,t)&=ig^{(S)}_{1o}{\tilde{s}}^{\alpha _o(t)}e^{\beta ^{(S)}_{1,o}\tau _o},\nonumber \\ O^{(D)}_{1}(s,t&)=ig^{(D)}_{1,o}\xi {\tilde{s}}^{\alpha _o(t)}e^{\beta ^{(D)}_{1,o}\tau _0},\nonumber \\ \tau _0&=3m_\pi ^2-\sqrt{9m_\pi ^2-t}. \end{aligned}$$

(A.2)

where ${\tilde{s}}=-i(s-2m^2)/2m^2$, $\xi =\ln ({\tilde{s}})$.

It is important that we use for single and double j-poles the pomeron and odderon intercepts equal to one

$$\begin{aligned} \alpha _p(t)=1+\alpha '_pt, \qquad \alpha _o(t)=1+\alpha '_ot. \end{aligned}$$

(A.3)

avoiding a violation of unitary restriction for total cross sections.

$$\begin{aligned}&P^{(S)}_{5}(s,t)=-g^{(S)}_{5,p}\dfrac{\sqrt{-t}}{2m}\lambda _+(t)\tilde{s}^{\alpha _p(t)}e^{\beta ^{(S)}_{5,p}\tau _p},\nonumber \\&P^{(D)}_{5}(s,t)=-g^{(D)}_{5,p}\dfrac{\sqrt{-t}}{2m}\lambda _+(t)\xi {\tilde{s}}^{\alpha _p(t)}e^{\beta ^{(D)}_{5,p}\tau _p}, \end{aligned}$$

(A.4)

$$\begin{aligned}&O^{(S)}_{5}(s,t)=ig^{(S)}_{5,o}\dfrac{\sqrt{-t}}{2m}\lambda _-(t)\tilde{s}^{\alpha _o(t)}e^{\beta ^{(S)}_{1,o}\tau _o},\nonumber \\&O^{(D)}_{5}(s,t)=ig^{(D)}_{5,o}\dfrac{\sqrt{-t}}{2m}\lambda _-(t)\xi {\tilde{s}}^{\alpha _o(t)}e^{\beta ^{(DS)}_{5,0}\tau _po} \end{aligned}$$

(A.5)

where m is proton mass. A choice of $\lambda _{\pm }(t)$ is discussed in the Sect. 3.1 .

The tripole terms $P^{(T)}, O^{(T)}$, containing only asymptotic main components from the Froissaron and Maximal Odderon [38]), which are defined in according with the AKM asymptotic theorem [42]

$$\begin{aligned}&P^{(T)}_1(s,t)=-g^{(T)}_{1,p}{\tilde{s}}\xi ^2 \dfrac{2J_1(z_p)}{z_p}e^{\beta ^{(T)}_{1,p}\tau _p},\nonumber \\&P^{(T)}_5(s,t)=-g^{(T)}_{5,p}\dfrac{\sqrt{-t}}{2m}\lambda _p(t)\tilde{s}\xi ^2\dfrac{2J_1(z_p)}{z_p}e^{\beta ^{(T)}_{5,p}\tau _p}, \end{aligned}$$

(A.6)

$$\begin{aligned}&O^{(T)}_1(s,t)=ig^{(T)}_{1,o}\tilde{s}\xi ^2\dfrac{2J_1(z_o)}{z_o}e^{\beta ^{(T)}_{1,o}\tau _o},\nonumber \\&O^{(T)}_5(s,t)=ig^{(T)}_{5,o}\dfrac{\sqrt{-t}}{2m}\lambda _o(t)\tilde{s}\xi ^2\dfrac{2J_1(z_o)}{z_o}e^{\beta ^{(T)}_{1,o}\tau _o}, \end{aligned}$$

(A.7)

where $z_p=r_p\tau \xi , \quad z_o=r_o\tau \xi , \quad \tau =\sqrt{-t/t_0}, \quad t_0=1\, \text {GeV}^2, \quad r_p, r_o$ are constants.

Contributions of the secondary reggeons, f and $\omega$ have a standard form

$$\begin{aligned}&R^{(f)}_1(s,t)=-g^{(f)}_{1}{\tilde{s}}^{\alpha _f(t)}e^{\beta ^{(f)}_{1}\tau _p}, \nonumber \\&R^{(f)}_5(s,t)=-g^{(f)}_{5}\dfrac{\sqrt{-t}}{2m}\lambda _p(t)\tilde{s}^{\alpha _f(t)}e^{\beta ^{(f)}_{5}\tau _p},\nonumber \\&\alpha _f(t)=\alpha _f(0)+\alpha '_f t,\end{aligned}$$

(A.8)

$$\begin{aligned}&R^{(\omega )}_1(s,t)=ig^{(\omega )}_{1}\tilde{s}^{\alpha _f(t)}e^{\beta ^{(\omega )}_{1}\tau _o},\nonumber \\&R^{(\omega )}_5(s,t)=ig^{(\omega )}_{5}\dfrac{\sqrt{-t}}{2m}\lambda _o(t) {\tilde{s}}^{\alpha _\omega (t)}e^{\beta ^{(\omega )}_{5}\tau _o}, \nonumber \\&\alpha _\omega (t)=\alpha _\omega (0)+\alpha '_\omega t. \end{aligned}$$

(A.9)

Appendix B: Original FMO model

We write here the explicit spin-non-flip terms of the FMO model at $t\le 0$ [38] just for reader’s convenience. The only difference of [38] is a notation for constants and functions. Spin-flip terms are presented and discussed in Sect. 3.2.

Froissaron and Maximal Odderon are written in the following form

$$\begin{aligned}&\begin{array} {ll} \dfrac{1}{iz}P^{(F)}_1(z_t,t)=h_{1,1}\xi ^2\dfrac{2J_{1}(r_{+}\tau \xi )}{r_{+}\tau \xi }e^{\beta ^{(F)}_{1,1}\tau _p} \\ \quad +h_{1,2}\xi \dfrac{\sin (r_{+}\tau \xi )}{r_+\tau \xi }e^{\beta ^{(F)}_{1,2}\tau _p} +h_{1,3}J_0(r_{+}\tau \xi )e^{\beta ^{(F)}_{1,3}\tau _p} ,\\ \end{array} \end{aligned}$$

(B.10)

$$\begin{aligned}&\begin{array}{ll} \dfrac{1}{z}O_1^{(M)}(z_t,t)= o_{1,1}\xi ^2\dfrac{2J_{1}(r_{-}\tau \xi )}{r_{-}\tau \xi }e^{\beta ^{(O)}_{1,1}\tau _o} \\ \quad + o_{1,2}\xi \dfrac{\sin (r_{-}\tau \xi )}{r_{-}\tau \xi }e^{\beta ^{(OM)}_{1,2}\tau _o} +o_{1,3}J_0(r_{-}\tau \xi )e^{\beta ^{(M)}_{1,3}\tau _o} ,\\ \end{array} \end{aligned}$$

(B.11)

where $z=2m^2z_t, \quad \xi =\ln (-iz_t), \quad \tau =\sqrt{-t/t_0}, \quad t_0=1\,\,\text {GeV}^2$, $r_- =r_+ -\delta r_-,\quad \delta r_-\ge 0$.

The effective secondary Regge pole contributions (crossing-even f and crossing-odd $\omega$) have the form

$$\begin{aligned} \begin{aligned}&{R_1^{(K)}}(z_t,t)=-\left( {\begin{array}{c}1\\ i\end{array}}\right) 2m^2g_1^{(K)}(-iz_t)^{\alpha _K(t)}e^{b_1^{(K)}t}\\&\alpha _K(t)=\alpha _K(0)+\alpha '_K t,\quad K=f,\omega \end{aligned} \end{aligned}$$

(B.12)

In order to perform calculations of the standard pomeron and odderon cuts in explicit analytical form we have made replacement $z_t\rightarrow z_0=z_t(t=0)$

$$\begin{aligned} \begin{aligned}&{R_1^{(L)}}(z_0,t)=-\left( {\begin{array}{c}1\\ i\end{array}}\right) 2m^2g_1^{(L)}(-iz_0)^{\alpha _L (t)}\\&\qquad \times \left[ d_Le^{b_{1,1}^{(L)}t}+(1-d_L) e^{b_{1,2}^{(L)}t}\right] , \\&\alpha _L(t)=1+\alpha '_L t, \quad L=P,O. \end{aligned} \end{aligned}$$

(B.13)

This parametrization takes into account a possibility of a non pure exponential behavior of the vertex functions for the standard pomeron and odderon [38].

The factor $2m^2$ is inserted in amplitudes $f,\omega , P,O$ in order to have the normalization for amplitudes and dimension of coupling constants (in milliibarns ) coinciding with those in [36].

We have added in the FMO the double pomeron and odderon cuts, PP, OO, PO in their usual standard form without any new parameters as well. Namely,

$$\begin{aligned}&P_1^{(2)}(z_t,t)=P_1^{(PP)}(z_t,t)+P_1^{(OO)}(z_t,t),\nonumber \\&O_1^{(2)}(z_t,t)=P_1^{(PO)}(z_t,t), \end{aligned}$$

(B.14)

$$\begin{aligned}&\begin{array}{ll} P_1^{(PP)}(z_t,t)&{}=-i\dfrac{2m^2(z_0g_1^{(P)})^2}{16\pi k s\sqrt{1-4m^2/s}} \left\{ \dfrac{d_p^2}{2B_1^p}\exp (tB_1^p/2)\right. \\ &{} +\dfrac{2d_p(1-d_p)}{B_1^p+B_2^p}\exp \left( t\dfrac{B_1^pB_2^p}{B_1^p+B_2^p} \right) \\ &{}\left. +\dfrac{(1-d_p)^2}{2B_2^p}\exp (tB_2^p/2) \right\} , \end{array} \end{aligned}$$

(B.15)

$$\begin{aligned}&\begin{array}{ll} P_1^{OO}(z_t,t)&{}=-i\dfrac{2m^2(z_0g_1^{(O)})^2}{16\pi k s\sqrt{1-4m^2/s}} \left\{ \dfrac{d_o^2}{2B_1^o}\exp (tB_1^o/2)\right. \\ &{}+\dfrac{2d_o(1-d_o)}{B_1^o+B_2^o}\exp \left( t\dfrac{B_1^oB_2^o}{B_1^o+B_2^o} \right) \\ &{}+\left. \dfrac{(1-d_o)^2}{2B_2^o}\exp (tB_2^o/2) \right\} , \end{array} \end{aligned}$$

(B.16)

$$\begin{aligned}&\begin{array}{ll} P_1^{PO}(z_t,t)&{}=\dfrac{2m^2z_t^2g_1^{(P)} g_1^{(O)}}{16\pi k s\sqrt{1-4m^2/s}}\\ &{}\times \left\{ \dfrac{d_pd_o}{B_1^p+B_1^o}\exp \left( t\dfrac{B_1^pB_1^o}{B_1^p+B_1^o}\right) \right. \\ &{}+ \dfrac{d_p(1-d_o)}{B_1^p+B_2^o}\exp \left( t\dfrac{B_1^pB_2^o}{B_1^p+B_2^o}\right) \\ &{} +\dfrac{(1-d_p)d_o}{B_2^p+B_1^o}\exp \left( t\dfrac{B_2^pB_1^o}{B_2^p+B_1^o}\right) \\ &{}+\left. \dfrac{(1-d_p)(1-d_o)}{B_2^p+B_2^o}\exp \left( t\dfrac{B_2^pB_2^o}{B_2^p+B_2^o}\right) \right\} \end{array} \end{aligned}$$

(B.17)

where $B_k^{p,o}=b_{1,k}^{P.O}+\alpha '_{P,0}\ln (-iz_0),\quad k=1,2 , \quad b_{1,k}^{P,O}$ are the constants from single pomeron and odderon contributions.

In [38] it was noted that for a better description of the data it is advisable to add to the amplitudes the contributions that mimic some properties of ”hard“ pomeron ($P^h$) and odderon ($O^h$). We take them in the simplest form

$$\begin{aligned} \begin{aligned} P_1^{(h)}(t)&=i2m^2z_t\dfrac{g_{h,p}}{(1-t/t_{p,h})^ 4},\\ O_1^{(h)}(t)&=2m^2z_t\dfrac{g_{h,o}}{(1-t/t_{o,h})^{4}}. \end{aligned} \end{aligned}$$

(B.18)