An upper bound on the total cross-section for electroweak baryon number violation

TL;DR

This paper establishes an upper limit on the cross-section of electroweak baryon number violation processes, suggesting potential observability at high-energy colliders and cosmic ray facilities, based on tunneling suppression bounds.

Contribution

It provides a new upper bound on the cross-section for electroweak baryon plus lepton number violating processes using recent tunneling suppression data and estimates of the pre-exponential factor.

Findings

Electroweak baryon violation processes may be observable at the LHC.

Potential detection at cosmic ray and neutrino observatories is not excluded.

Higher partial waves could contribute significantly to observable rates.

Abstract

An upper bound on the total cross-section of s-wave electroweak instanton/sphaleron induced baryon plus lepton number violating processes is presented. It is obtained by exploiting a recently reported lower bound on the corresponding tunneling suppression exponent and by estimating the pre-exponential factor. We find that the present knowledge about electroweak baryon plus lepton number violating processes still allows their eventual observability at the Very Large Hadron Collider, even as pure s-wave scattering. A possibly observable rate at cosmic ray facilities and neutrino telescopes is presently not excluded, but requires a substantial contribution from higher partial waves.

Equations

$${\hat{\sigma }}_{\mathrm{ff}}^{(I)}$$

$$F_W(ϵ)=1-\frac{3^{4/3}}{2}{ϵ}^{4/3}+\frac{3}{2}{ϵ}^2+\mathcal{O}({ϵ}^{8/3}).$$

$$P(\sqrt{\hat{s}},n)\propto \exp \left[-\frac{4\pi }{{\alpha }_W}{F}_W\left(ϵ,\nu \right)\right],$$

$$F_W\left(ϵ\right)=\underset{\nu \to 0}{\lim }{F}_W\left(ϵ,\nu \right).$$

$${\hat{\sigma }}_{\mathrm{inel}}^{(\ell =0)}<\frac{4\pi }{\hat{s}}\simeq 5.5\mathrm{pb}\frac{1}{{ϵ}^2},$$

Full text

DESY 03-076

hep-ph/0307034

An upper bound on the total cross-section for electroweak baryon number violation

A. Ringwald

Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany

An upper bound on the total cross-section of s-wave electroweak instanton/sphaleron induced baryon plus lepton number violating processes is presented. It is obtained by exploiting a recently reported lower bound on the corresponding tunneling suppression exponent and by estimating the pre-exponential factor. We find that the present knowledge about electroweak baryon plus lepton number violating processes still allows their eventual observability at the Very Large Hadron Collider, even as pure s-wave scattering. A possibly observable rate at cosmic ray facilities and neutrino telescopes is presently not excluded, but requires a substantial contribution from higher partial waves.

1. Baryon and lepton number are not strictly conserved in the standard electroweak model [1]. In the background of topological non-trivial fluctuations of $SU(2)$ gauge fields with topological charge $Q\not=0$, notably of instantons ($Q=1$) and anti-instantons ($Q=-1$) [2], baryon ($B$) and lepton ($L$) number change according to $\triangle B=\triangle L=-n_{\rm gen}\,Q$, where $n_{\rm gen}=3$ is the number of fermion generations. Such fluctuations correspond to transitions between degenerate, topologically inequivalent vacua [3]. The latter are known to be separated by energy barriers whose minimum height is given by the static energy $E_{\rm sp}$ of an unstable static solution of the classical Yang-Mills equations called the sphaleron [4]. Its value111Here, $m_{W}=80.423(39)$ GeV is the W± boson mass and $\alpha_{W}(m_{W})=0.033819(23)$ is the $SU(2)$ fine structure constant [5]., $E_{\rm sp}\approx\pi\,m_{W}/\alpha_{W}\simeq 7.5$ TeV, sets the scale for non-perturbative $B+L$ violation in the Standard Model. Indeed, while these processes are exponentially suppressed at energies or temperatures below the sphaleron energy by a tunneling factor, they are known to have a sizeable rate for temperatures above $E_{\rm sp}$ [6] and to have a crucial impact on the evolution of the baryon asymmetry of the universe. On the other hand, the long-standing question, first raised in Refs. [7, 8], whether these processes occur with sizeable rates in high energy particle collisions, is still not finally settled (for reviews, see Ref. [9]).

It is the purpose of this Letter to elaborate and to compare recent results on this issue [10, 11, 12] and to work out an upper bound on the total cross-section for electroweak baryon number violating processes. We will show that the present knowledge about the latter still allows their eventual observability at future colliders [11, 13], such as the Very Large Hadron Collider (VLHC) [14]. We will also discuss implications for searches at cosmic ray facilities and neutrino telescopes [15, 16].

2. At center-of-mass (CM) energies much less than the sphaleron energy, the rates of anomalous electroweak $B+L$ violation are rapidly growing [8]. The corresponding total cross-section is known to have an exponential form [17]. Including essential pre-exponential factors [18], one has, for the phenomenologically interesting case of fermion-fermion scattering via electroweak instantons/sphalerons, ${\rm f+f}\stackrel{{\scriptstyle I}}{{\to}}{\rm all}$,

[TABLE]

where $\sqrt{\hat{s}}$ denotes the fermion-fermion CM energy. By means of perturbative calculations of the relevant exclusive amplitudes about the instanton ($I$), squaring them and summing over the final states, or, alternatively, by means of a perturbative calculation of the forward elastic scattering amplitude about the widely separated instanton anti-instanton ($I\overline{I}$) pair and determining the imaginary part to get the total cross-section via the optical theorem, one may calculate the decisive tunneling suppression exponent $F_{W}$, which is sometimes called “holy-grail function” [9], as a series in fractional powers of $\epsilon\equiv\sqrt{\hat{s}}/(4\pi m_{W}/\alpha_{W})\simeq\sqrt{\hat{s}}/(30\ {\rm TeV})$ [18, 19],

[TABLE]

Correspondingly, the total cross-section (1) is exponentially growing at $\epsilon\ll 1$. At $\epsilon={\mathcal{O}}(1)$, however, the perturbative expression (2) ceases to be helpful. In this energy regime, only estimates/extrapolations of and lower bounds on the tunneling suppression exponent exist (cf. Fig. 1 (top)), which have been quantified recently [10, 11, 12].

3. In almost all theoretical investigations of electroweak $B+L$ violation in high energy collisions, the pre-exponential factor in Eq. (1) is not considered. It is, however, numerically very large and has therefore to be taken into account before any conclusion on the observability of the effect can be drawn. Its size is mainly determined by the large universal factor $(2\pi/\alpha_{W})^{7/2}\simeq 8.7\times 10^{7}$, where the exponent $7/2$ can be easily understood as being the result of subtracting from the familiar nominal power $4\,N_{c}=8$ [1] the power $9/2$ – nine being the effective number of collective coordinate saddle-point integrations [18, 11]. The large pre-exponential factor in Eq. (1) leads to the fact that a cross-section \hat{\sigma}^{(I)}_{\rm ff}\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,10^{-3} fb, observable at the projected VLHC222The VLHC has a projected proton-proton CM energy of $\sqrt{s}=200$ TeV and a luminosity of about ${\mathcal{L}}\approx 6\cdot 10^{2}$ fb*-1* yr*-1* [14]. [14], requires only a moderate reduction in the tunneling suppression exponent, F_{W}\mbox{\,\raisebox{1.29167pt}{<}!!!!!\raisebox{-3.87495pt}{\sim}\,}0.12, from its value at zero energy, $F_{W}(0)=1$. If a sizeable reduction in F_{W}\mbox{\,\raisebox{1.29167pt}{<}!!!!!\raisebox{-3.87495pt}{\sim}\,}0.02 is realized in nature, it is even possible to obtain a large cross-section of hadronic size, \hat{\sigma}^{(I)}_{\rm ff}\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,1 mb, which is observable in present day or near future cosmic ray facilities and neutrino telescopes in the form of cosmic proton or cosmic neutrino initiated events [15, 16].

This numerical fact is clearly demonstrated by the result for $\hat{\sigma}^{(I)}_{\rm ff}$ obtained in Ref. [11] (cf. Fig. 1 (bottom; solid and dotted)), where the tunneling suppression exponent has been estimated via the optical theorem from the $I\overline{I}$ interaction, known in pure $SU(N_{c})$ gauge theory for arbitrary separation [10, 20], (cf. Fig. 1 (top; solid)), and where the complete pre-exponential factor, with inclusion of its energy dependence, has been calculated in the saddle-point approximation. First of all, we see that the approximation of the pre-exponential factor displayed in Eq. (1), leading to the estimate shown as a dotted line in Fig. 1 (bottom), gives in fact a reliable estimate of the complete pre-exponential factor over almost the full energy range. Moreover, the estimate of the cross-section becomes indeed of order $10^{-3}$ fb ($1$ mb) as soon as the estimate of the tunneling suppression exponent becomes of order $0.12$ ($0.02$).

The analytical estimate based on the $I\overline{I}$ interaction in pure gauge theory is based on a number of assumptions which, at energies above the sphaleron, \epsilon\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,0.3, may or may not be valid. Another method to infer the tunneling suppression exponent was proposed in Refs. [21, 22, 23] and the results of a respective extended quantitative study was presented recently in Ref. [12]. The method is based on the observation that the inclusive probability $P(\sqrt{\hat{s}},n)$ of tunneling from a state with energy $\sqrt{\hat{s}}$ and number of incoming particles $n$ is calculable semi-classically, provided that $\epsilon\equiv\sqrt{\hat{s}}/(4\pi m_{W}/\alpha_{W})$ and $\nu\equiv n\,\alpha_{W}$ are held fixed in the limit $\alpha_{W}\to 0$. In this regime, the probability $P(\sqrt{\hat{s}},n)$ has also an exponential form,

[TABLE]

and the corresponding tunneling suppression exponent $F_{W}(\epsilon,\nu)$ can be obtained by solving a classical boundary value problem for the Yang-Mills equations. Furthermore, it has been conjectured – and proven in the first few orders of the perturbative expansion in fractional powers of $\epsilon$ (cf. Eq. (2)) [22, 24] as well as by comparison with the full quantum mechanical solution in a model with two degrees of freedom [25] – that the tunneling suppression exponent for the two-particle cross-section is recovered in the limit of a small number of incoming particles,

[TABLE]

In Ref. [12], the relevant classical boundary value problem was solved, for spatially spherically symmetric333Without this simplification provided by spherical symmetry, the computational cost of the numerical analysis seems to be prohibitive at present. configurations of the $SU(2)$-Higgs gauge theory, for a large range of $\epsilon$ and $\nu$. Though computational limitations did not allow to reach literally the limit $\nu=0$, the authors were able to extrapolate their results for the tunneling suppression exponent for multiple incoming particles, $F_{W}\left(\epsilon,\nu\right)$, to zero $\nu$ and to get thereby a stringent lower bound on $F_{W}(\epsilon)$ (cf. Fig. 1 (top; dashed shaded)). In addition, they provided an independent estimate for $F_{W}(\epsilon)$ (cf. Fig. 1 (top; dashed)).

It is important to note that the lower bound on and the estimate of $F_{W}(\epsilon)$ obtained in Ref. [12] are applicable, strictly speaking, only for s-wave scattering, since they have been obtained exploiting spatially spherically symmetric configurations. This property is shared by the dominant intermediate saddle-point configurations encountered in the $I\overline{I}$ “valley” configuration [10, 20]. Therefore, the estimate of $F_{W}(\epsilon)$ from Ref. [12] should, at small energies, coincide with the analytical estimate based on the $I\overline{I}$ interaction. This is clearly demonstrated in Fig. 1 (top; dashed and solid, respectively). At higher energies, above the sphaleron, \epsilon\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,0.3, this coincidence is no more observed. Apparently, the estimate based on the $I\overline{I}$ interaction, which has been argued in Ref. [11] to be reliable for \epsilon\mbox{\,\raisebox{1.29167pt}{<}!!!!!\raisebox{-3.87495pt}{\sim}\,}0.75, gives a more optimistic value for the two-particle tunneling suppression exponent than the one from the extrapolation of $F_{W}\left(\epsilon,\nu\right)$ to $\nu=0$.

4. At energies above the sphaleron, \epsilon\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,0.3, the total cross-section of electroweak baryon number violation may well be dominated by axial symmetric configurations [26], on which there is at present no information available. Nevertheless, it is very instructive to exploit the lower bound on the tunneling suppression exponent for s-wave scattering from Ref. [12] and to determine from it, via Eq. (1), an upper bound on the cross-section of s-wave electroweak $B+L$ violating processes initiated by the scattering of two fermions. We find that the estimate based on the $I\overline{I}$ interaction violates this s-wave upper bound for \epsilon\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,0.9 (cf. Fig. 1 (bottom)). However, as shown in Fig. 1 (bottom; dashed shaded), the upper bound is rapidly getting less stringent and exceeds \mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,10^{-3} fb for \epsilon\simeq\sqrt{\hat{s}}/(30\ {\rm TeV})\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,1.2. Therefore, it does not exclude the observability of electroweak $B+L$ violation at the VLHC. The upper bound even grows above $1$ mb ($100$ mb) for \epsilon\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,7.5 (\epsilon\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,9.7), thus still allowing for observable consequences of anomalous electroweak processes in ultrahigh energy (E=\hat{s}/(2\,m_{p})\,\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,\,5\times 10^{19} eV) cosmic ray and neutrino physics [15, 16]. However, in order to reach such an ${\mathcal{O}}({\rm mb})$ cross-section, as also suggested by the $I\overline{I}$ estimate in Fig. 1 (bottom), a substantial contribution from higher partial waves is required. The $I\overline{I}$ estimate and the upper bound violate the s-wave unitarity bound for the inelastic cross-section,

[TABLE]

above \epsilon\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,1.1 and \epsilon\mbox{\,\raisebox{1.29167pt}{>}!!!!!\raisebox{-3.87495pt}{\sim}}\,1.9, respectively.

Acknowledgements

I would like to thank F. Bezrukov, V. V. Khoze, V. Rubakov, P. Tinyakov, and F. Schrempp for fruitful discussions, helpful comments, and a careful reading of the manuscript. I would also like to thank the authors of Ref. [12] for providing their data concerning the tunneling suppression exponent.